Hilbert space has no physical extent or extent in time. Quantum wave functions live in Hilbert space.
It really is like Einstein said: if you think you understand quantum mechanics then you just haven't been paying attention enough!
Quantum mechanical wave functions are weirder than anyone thought they were and they were plenty weird before. Einstein, et al. defined how physical theories work, they consider locality critical: Two particles interact by being in the same place at the same time. There is no 'action at a distance' in modern physics. Things happen by exchanging particles. The fastest these particles can travel is at the speed of light. That's why special and general relativity produce such weird effects, nothing can go faster than the speed of light.
That assumption just doesn't hold in quantum mechanics or quantum field theories. Well, not quite. In addition to wave functions that represent particles that effectively can't go faster than the speed of light, there's some wiggle room on that. Wave functions only predict the probability distribution of making a particular measurement. You can measure a particle going faster than the speed of light, you just won't know where the particle is. This is Heisenberg's uncertainty principle. So particles sometimes go faster than the speed of light, but for a short time (a very short time) and it has consequences. This is a rare event and no information is transmitted faster than the speed of light, the knowledge of the speed is traded for lack of knowledge of the particles whereabouts. It all balances out. So every equation that physicists believe to be true depend upon locality: except for quantum mechanics. The standard model for quantum mechanics says that you can change Hilbert space, one parameter in Hilbert space, and every single wave function in the universe will recognize that change instantaneously, and if you configure it correctly the change can actually cause an effect on a wave function at an earlier time! The measurement can change the distribution of measurements that you have already recorded but not looked at. That's right. I said it. It says information can travel backwards in time.
You can make a measurement that will affect a separate measurement made in the past. Think about that. If that's not time travel, then what is? Of course this can't be used to send information into the past faster than the speed of light... (DAMN!) It only changes the distribution of measurements, you have to know the results of the future measurements to verify the current measurements were altered, it's only the correlations of the two measurements that changes.
Like Einstein said about the neutrino: "Who ordered that?"
Einstein went on with Bell and Rosen and Podolsky to propose the measurements above in the hopes that they would show that quantum mechanics was actually a 'hidden variable' theory and nothing in the theory changed faster than the speed of light, but to their chagrin, the experiments proved the exact opposite. They showed that modifications to Hilbert space, unlike those to the space we can see that are limited to propagating at the speed of light, happen instantaneously across the entire universe. Hilbert space, where quantum mechanical wave functions live, is different than our observed space and time. Changes made in Hilbert Space can have effects on the past, can travel faster than light and do not happen in the real world, but they do affect it.
How has science proven this? It seems just phantasmagorical! It's all about the fundamental tenets of quantum mechanics...
Quantum mechanics has this interesting feature called entanglement: what does this actually mean? It means that when you make measurements in the real world you change the state of the wave function every where in Hilbert Space instantaneously and hence the measurement you make in the real world is also altered. It basically says that quantum mechanics is a theory of a linear superposition of quantum states in Hilbert space. The way to affect the coefficients of the superposition (how much of this and how much of that) is to make a measurement. And these effects are manifested across the universe instantaneously, even having effects back in time if they have to.
When you make a measurement you get one bit out of the measurement. The most illustrative example is measuring the spin of an electron. Spin can be thought of as the rotation of the election (not really, since the electron is a point particle as far as we can tell, so there's nothing there to actually spin... except the wave function can spread the electron around in space and that allows for the effects of spin to show up. An electron can only be in a single state after it is measured. That is what information is. It tells you if the wave functions (q-bits) are in a given state or not (can you measure the state?))
One of the most interesting things about quantum mechanics is the 'no-cloning' proposition. In order to preserve information (which we assume is a conserved quantity: is neither created nor destroyed) you cannot clone a quantum state without destroying it. In other words, you can copy a particular quantum state, but then you don't know what state the original system is in now. Without this proposition, you can send information faster than the speed of light. Einstein had prophesied in special relativity that no information could be transported faster than the speed of light. So far, he's been shown to be right. If information is conserved then the no-cloning theory of quantum mechanics is a consequence of this conservation law.
When something is preserved it points to a symmetry. Symmetries are very important to physics. Symmetry under rotation leads to conservation of angular momentum. Symmetry under translation leads to conservation of energy. Symmetry under time and charge leads to conservation of parity. Although all three symmetries interact to cause one symmetry to be preserved. So preservation of information, a grand symmetry if there ever was one, must be able to be shown to hold for quantum mechanical equations. And Lo and Behold: it does! What symmetry leads to conservation of information? The current claim is that they symmetry of equations in time combined with the purity of quantum states leads to the conservations of information The universe is all about information.  What does the purity of a quantum state mean? It means it's an eigenstate in Hilbert space. That's like being identified with an axis on a graph. Any state in Hilbert space has only one representation from a given set of quantum eigenstates. In other words, Hilbert space is linear and states are made up of linear superpositions of eigenstates. Think of it as one eigenstate cannot be made up of a superposition of any of the other eigenstates, they are all perpendicular to each other.
So somehow the universe changes the Hilbert space the wave function lives in instantaneously across the entire universe. Observable or not. It only requires that the two q-bits be entangled sometime in the past (and everything has been entangled since the big bang.) This preserves the transmission of information to the speed of light still. Even though you have changed the probability of the measurement on the receiving side instantaneously you can only tell that is true if you get the information about the measurement from the transmission side which can only be sent at the speed of light (Oops, no information can be transmitted faster than the speed of light, as far as we know) and information needs to be recorded on a physical thing (outside of Hilbert space) and physical objects are limited to traveling at the speed of light. So no contradictions from quantum mechanics are predicted when integrating with general relativity.
Okay, that's weird enough. Wave functions are not in the real world, the physical world, they live in Hilbert space. You are a quantum mechanical wave function, so you really live in Hilbert space. All quantum mechanical wave functions live in Hilbert space. What does this mean?
[Side bar: Hilbert was the mathematician that posed the ten most interesting mathematical problems to solve to finish the entire description of mathematics.  All of them have not been solved yet, although the solutions have broken the idea that the answers to these problem would solve mathematics forever. Only three of his proposed problems remains unsolved (some of them were not formulated carefully enough but have been solved for the different formulations), but the most famous (and most useful) problem to remain unsolved is Reiman's conjecture about the zeros of the Riemann equation and the distributions of primes (Reiman's theorem is amazing! He proposed an equation that had a solution of zero for every prime number. This is super awesome, as now there is an analytical method to describe the distribution of primes. Assuming the proposition is true, of course...) End of Sidebar]
[Sidebar: Hilbert didn't like some of the answers he got to his top 24 problems. Actually he probably did like them, but they totally eradicated his original proposal that mathematics could be solved. Gödel, Turing and Church killed that idea with a vengeance! They showed that any system as complicated as the integers was too complicated to prove every truthful statement that could be written or even more germane: if you could prove every truthful statement then the theory was inconsistent (you can prove false things.) Which answer do you pick? Wholly shit! Of course in any sane universe you pick the former: it's consistent. But there are things that are true that you can never prove to be true and there are things that are false that you can never prove are false. But every statement you can prove is either true or false, it can't be both. This is a much better choice than inconsistency. If a system is inconsistent then it is useless. It can't be used to predict anything. this is the one thing I have faith in. Otherwise the entire world is less complicated than the integers. Hardly seems likely! End of sidebar]
So let's summarize the current state of Einstein's universe: Physical objects can't travel faster than the speed of light. Physical objects have mass and energy. These two things are equivalent and can change into each other. The interactions of masses are governed by Einstein's laws: No matter how fast you are travelling relative to another frame of reference, when you measure the speed of light it's always the same. How did Einstein figure out that mass and energy are the same thing? It's a consequence of his equations: If something has a mass and energy in one frame these quantities are added together to predict it's motion. They transform together to a new frame. The equation treats them as the same thing. He figured out the conversion constant from his transformation equations:
E = mc^2.
This is weird, if light (and the other forces) were disturbances in space you would expect that if you travelled faster towards or away from a beam of light it would move faster or slower, but it doesn't. This means that as you speed up you bend space around you. It's like it can't keep up to the change in information at the speed of light. And it can't, you can only effect space that is close enough to you in time and space (this is the assumption of locality plus the universality of the speed of light.) This is the same as saying that a piece of mass or a q-bit after it's measured is a physical thing. The measurement tells you if it is in a particular state or not, that's one bit of information. Before the measurement the q-bits are entangled and in a particular superposition of states. The coefficients change as a function of time. Simultaneously (correlated) in the entire universe.
As discussed previously, this does not allow you to send information faster than the speed of light. The particle has to get there, and it is subject to Einstein's laws of general relativity. In fact, we made sure quantum mechanics was consistent with relativity and changed its formulation to make it so.
There are two possibilities: either the wave function changes instantaneously across the entire universe or the q-bits didn't really go into a superposition of states they just went into one particular state and evolved over time from there. No surprise. In fact, Einstein was a big fan of the second theory, it's called the theory of hidden variables. He and Rosen and Bell  devised an experiment to tell the difference between the two hypothesis. Einstein was very disappointed to learn that the first theory was correct.
But it's about to get weirder than that.
Our new understanding of quantum mechanics says that the change in the wave function does not change simultaneously all across the universe. It says that the effects of the change of the wave function can propagate back in time in the real world. Yes, that's right. I make a measurement now. You made a measurement in the past. I didn't change the measurement you made but I changed the distribution you measured it out of. So if you make this measurement many many times, it will be different if I make a measurement afterwards than if I do not make the measurement. I can change the past. Quantum mechanical theory insists that this is true. Does Hilbert space have time? It has a Newtonian notion of time, it has to or it won't work. Unlike real space, time and the eigenstates don't mix. In the real world Einstein showed that they do mix. They can't mix in Hilbert space as this would ruin the purity of the eigenstates. 
You can affect the past. By making a measurement. By making certain measurements now, I can change what you will measure back then. My head hurts.
WTF? How does this make any sense?
[Sidebar: The other explanation proposed is that the q-bits were not in a superposition of states but in a particular state. This is the hidden variable theory. Why? Because there is some distribution of values that told the qubits to pick a particular state back in the last time you tried to entangle them. It turns out that the Bell theorem based on the Einstein-Rosen-Podolsky conjecture  tells you how to distinguish these two options. Lo and behold: quantum mechanics is correct, hidden variable theories are not correct. Hilbert spaced changes instantaneously across the entire universe. End sidebar]
Putting the q-bit in a particular state (not a superposition of states) doesn't work. If this were true you would measure a certain distribution at the receiver. If the wave function changes everywhere at the same time in the universe you measure a different distribution. We've done the experiment (proposed by Bell following Einstein, Rosen and Podolsky) thousands of times. The answer is always the wave function in Hilbert space changes instantaneously. The answer is never that it chooses a particular state at the time of entanglement.
What does this say? It says the wave function is not in physical space but in Hilbert space. What does that mean? It just means the quantum wave function really is a linear superposition of Hilbert eigenstates. There are other ways to test this: they all give the same answer. Wave functions come in linear superpositions of states and you can make measurements that tell you what state the q-bit is in. The state of all entangled q-bits (q-bits that are superimposed with each other) also change state at the same time across the universe, if one q-bit in the entanglement is measured..
This measurement gives you one bit of information. This is why the state has to change simultaneously across the inverse. To preserve information you must know what the measurement distribution will be in the second measurement as you measured a q-bit in a correlated state with the original q-bit. Of course they are correlated, they are directly related, there's only one bit of information measured here . If the wave function did not change instantaneously across the universe information would not be preserved.
There must be some equations (symmetries) that the conservation of information predicts. What are they? Are they quantum mechanics? Is it the purity of the linear superposition of quantum states? Time translation symmetry predicts energy conservation. The conservation of informations is the result of cause and effect always being related by locality. This is the basics of classical physics, but for quantum mechanics this does not hold; so making predictions from quantum mechanical theory is really hard because it is like nothing that we know of in the real world, even though it's the firmament of the real world.
thanks for reading.
 https://arxiv.org/pdf/1010.5300.pdf where quantum mechanics is derived from information theoretical considerations, for instance. While they do not discuss the conservation of information directly https://medium.com/the-physics-arxiv-blog/how-quantum-mechanics-derives-from-a-revolutionary-new-theory-of-information-4487489dbb34 I believe that this assumption can lead to their postulates. Also discussed in: https://arxiv.org/abs/1208.0493. But the real claim is shown from https://philarchive.org/archive/CHICOI, however, in this paper they insist on the "no time travel in real space" which seems contradictory to the main discussion here. This is only apparent. We are talking about changing distributions in the past, not measurements.
 Actually turned into 23 or 27 problems: https://en.wikipedia.org/wiki/Hilbert%27s_problems
 An interesting question: what would be the consequences to quantum theory if time in Hilbert space was like Einsteins time in real space? Are the effects just to small to be noticed?
 Einstein-Rosen-Padolsky conjecture: https://en.wikipedia.org/wiki/EPR_paradox and https://en.wikipedia.org/wiki/Bell%27s_theorem