Is there such a thing as "Action at a distance"? What ever happened to "action at a distance" in entangled quantum states, i.e. the Einstein-Rosen-Podolsky (EPR) paradox? I thought they argued that in principle one could communicate faster than speed of light with entangled, separated states when one wave function gets collapsed. I imagine this paradox has been resolved, but I don't know the reference.
 A: Well, the problem in that paradox is that yes, one of the parties will measure the entangled particle to get the wave function collapsed and yes it will collapse for the other party. However, the other party will still have to measure the thing to learn what it is or has to wait for the initial party to send them a message telling what the wave function has collapsed to. The first method will result in a 50% +x and 50% -x(if it is spin you are measuring), as the wave function that collapsed can collapse to either one of these states. So the fact that the wave function collapsed does not really transfer any usable information to the other side. The second method is capped with the speed of light anyway.
A: Let's be more rigorous. No-signalling has been proven safely and shouldn't be worried about. Nevertheless, you'd notice that the point of EPR paper was to show that if quantum mechanics is considered to be a description of "reality", then it is "incomplete". There is an approach, such as in operationalism, to say quantum mechanics isn't meant to be a description of reality. It's a description of our knowledge of reality, due to Asher Peres. Another approach is to say we can give an ontological model of quantum theory using contextual hidden variables, such as the one in de Broglie-Bohm model.
So conclusion: EPR argument hasn't been resolved if you mean it's gone! Because in fact orthodox quantum mechanics isn't a complete description of reality. However, it doesn't mean one can signal faster than light!
Some interesting extra information: There is an interesting paper which kind of analyses Einstein's argument. It bring historical facts that show Einstein didn't like the EPR and wrote another paper with the same title in correspondence with Schrödinger, and the one with Rosen and Podolsky was never reviewed by him. 
This quotation is from a letter of Einstein to Schrödinger, dated June 19, 1935:

“For reasons of language this [paper] was written by Podolsky after many discussions. But still it has not come out as well as I really wanted; on the contrary, the main point was, so to speak, buried by the erudition.”

Update: A source of confusion in my answer has been pointed out by Marek. I'll try to clarify here:
Scientific realism assumes there is an underlying objective reality which has attributes regardless of them being measured by an observer. 
One can suggest a model which ignores such reality and say "...there is no logical necessity for a realistic worldview to always be obtainable"(Fuchs and Peres, Physics Today 53 (3), 70-71.).
On the other hand, one can offer an ontological model which in this case it can be located in 3 different category as in figure below:
 
I believe Einstein had in mind to show that quantum mechanics can't give a picture of type (a). Which was successful. Because, even if there is an underlying reality, quantum states can't sharply specify them by any means. 
A: It's not possible to communicate faster than light using entangled states. All you get out of entanglement is a correlation between the values of two measurements.; the entanglement doesn't allow you to influence the value measured at another location in a non-causal way. In other words, the correlation only becomes evident after combining the results from the measurements afterwards, for which you need classical information transfer.
For example, consider the thought experiment described on the Wikipedia page for the EPR paradox: a neutral pion decays into an electron and a positron, emitting them in opposite directions and with opposite spins. However, the actual value of the spin is undetermined, so with respect to a spin measurement along a chosen axis, the electron and positron are in the state
$$\frac{1}{\sqrt{2}}\left(|e^+ \uparrow\rangle|e^- \downarrow\rangle + |e^+ \downarrow\rangle|e^- \uparrow\rangle\right)$$
Suppose you measure the spin of the positron along this chosen axis. If you measure $\uparrow$, then the state will collapse to $|e^+ \uparrow\rangle|e^- \downarrow\rangle$, which determines that the spin of the electron must be $\downarrow$; and vice versa. So if you and the other person (who is measuring the electron spin) get together and compare measurements afterwards, you'll always find that you've made opposite measurements for the spins. But there is no way to control which value you measure for the spin of the positron, which is what you'd need to do to send information. As long as the other person doesn't know what the result of your measurement is, he can't attach any informational value to either result for his measurement.
A: One of the fundamental tenets of relativity is that all observers are correct. Two people pass each other near the speed of light and they will each think that the other person's clock is running more slowly than their own. Who's right? They both are. It doesn't matter if you have to bend and warp and crumple the universe itself in order to make them both correct, and in fact that is exactly what Einstein did with general relativity. Now if you have a photon moving at the speed of light, it observes infinite longitudinal length contraction, meaning it sees the universe as an infinitely thin sheet of paper that it is passing through. From its own perspective, it is created and absorbed at a single point in space, all during a single moment of time. What we perceive as spatial extent and temporal duration of the photon's path are both, and only, consequences of relativistic dilation from our own perspective. Now consider three photons having mutually orthogonal trajectories. Each thinks the universe is infinitely thin in its direction of motion, and each traverses that universe in a single instant. They must all be right. This happens only if the universe is itself only a single point which lasts only for a single moment. (If this last conclusion is troubling, it might be helpful to presume that the mutually orthogonal entangled emissions arise together from a single event.)
Now comes the Einstein-Podolsky-Rosen paradox: how can two particles, which themselves have spins in absolutely as-yet undefined directions, after traveling light-years apart, "decide" that the instant one of them is measured, thus collapsing itself into a totally random distinct spin state, then the other one, no matter how far away, and no matter how much time later, will always, once measured, collapse into the opposite spin state in order to preserve total angular momentum? Einstein called this "spooky action at a distance" and believed that it couldn't happen -- that the particles, which cannot communicate with each other (else that information would have to impossibly travel faster than light), must have some "hidden variable" inside them which determines which spin state will arise, once measured. However, the Bell Inequality, which I won't describe here, proves inescapably that there cannot be an Einstein "hidden variable" making the determination. What then are we to do? Even if we have to bend and warp and crumple the universe itself, Einstein and Bell must both be right. And that is exactly what we have to do. The particles do not engage in "spooky action at a distance" because they are both at the same point in space, and at the same moment in time, when they are measured. That's because the universe in which they reside is itself a single point in space and a single moment in time. This is a little different from the "non-locality" of the universe asserted in some literature. Here, the universe is entirely local, but all at a single point of spacetime. This then has been just one person's personal perspective on the EPR paradox, but something similar has been said before: a thousand years is like unto a day, and the heavens can be measured in the palm of the hand.
Michael C. Kleder, 7 Feb 2021
A: Being silly, in the vein of the season, if you agreed before hand that if in David's example your colleague could open their christmas present on the other side of the world if their electron was spin up, and the same time they measured their electron you measured your positron as spin down, you now know they have opened their present, faster than the speed of light
Instantaneous data transfer!
A: Everybody misses the point Einstein was trying to make, which makes it all the more remarkable that it's been 80 years since he was working on spooky action at a distance. The no signalling theorems mean nothing, and it's a shame that most answers simply site: no signalling, nothing spooky about it. Bell emphasized that his theorem could be quickly summarized as: there is non-locality. Guess what? Bell was very well aware of the no signalling theorems. The point is not that we can send signals, the point is that there is a signal sent by the photons themselves -- they have to be communicating. How else could they always coordinate their spins? Einstein's whole critique of quantum mechanics was that it needed to be like Bertleman's socks -- e.g. that the spins were already determined before the experiment, or else there would have to be a non-local communication to coordinate the spins. Einstein called it telepathy, and it's been proven by Bell. 
If you don't think there's spooky action, then how do you explain that the spins are always coordinated? If you gave Alice and Bob each a quarter and separated them by a large distance and tasked them with choosing heads or tells, and they always came back to you with one choosing heads, the other choosing tails, what might you think? Maybe they talked to each other on a phone and coordinated their results? 
