(Sorry for replying so late, I hope you'll still see this.)
Well, the situation of entanglement is not really* different from the following one:
Imagine you have two pairs of socks, a red pair and a blue pair. Now, certainly you only wear a red (blue) sock on your left foot iff you wear one of the same color on your right foot, right?
So, consider the measurement of your left sock's color by pulling up the leg of your pants and looking at your sock. If it is red, you can be 100% sure you will measure your right sock to be "red" as well, and, similarly, if it is blue you will measure blue on the right, too. Now, in order to explain this phenomenon, do you need faster-than-light travel of your left sock's color to your right sock at any point? No. The results of the color measurements on both sides are just correlated with each other because both subsystems (both socks) were prepared that way earlier.
* Granted, the issue of entanglement is actually slightly more complicated since a sock being either red or blue is not quite the same as the sock being in a superposition of both colors, i.e. being red and blue at the same time. But this difference just boils down to the fact that, in (the Copenhagen interpretation of) quantum mechanics there are measurements whose outcome is non-deterministic and superpositions just encode this very fact (together with complementary observables that allow us to distinguish between pure eigenstates and superpositions in the first place. Without complementary variables we would never see a difference between the two.) So, for instance, in entanglement experiments one often prepares two particles in a superposition of eigenstates of the spin-z operator and then proceeds to measure the particles' spin with respect to another axis. It turns out that the results of both measurements will be random but correlated. Will you able to explain the results using the sock analogy? No, because the socks are entirely classical, in particular deterministic, and there are no complementary variables such as spin x and spin z (e.g. the color of the socks has nothing to do with it being made from cotton or wool and both properties are defined independently). Will you now need non-locality, though? Still no, you just needed to add non-determinism in the form of complementary variables to your theory.
So, put differently, if your theory of quantum mechanics is non-deterministic, then it can very well be local. Only if it is deterministic, it must be non-local (see, for instance, Bohmian mechanics). So, while you could have both, indeterminism and non-locality, at the same time, you certainly don't need both and, in fact, in the standard (Copenhagen) interpretation of quantum mechanics we don't have both. I think the missunderstanding that the Copenhagen interpretation is both non-deterministic and non-local originates from the idea that we imagine the wave function to collapse upon measurement and this collapse to propagate faster-than-light from one entangled particle to the other. But the collapse of the wave function is not just not compatible with special relativity but it is also not compatible with the unitary time evolution of quantum mechanics, so we should consider it merely a tool for us instead of something that is actually happening. What matters at the end of the day are expectation values of measurements and for that you don't need the collapse of the wave function or non-locality at any point.
Your example has the problem of having a 'hidden' variable; the assignment of sock to foot. When you put on your socks, you assign a sock to a foot. So, when you later look, this determines the outcome, in this case the color of your socks.
This is not the case with the classical Copenhagen interpretation of QM - there are no hidden variables. I.e., the assignment of sock to foot happens at the moment when you look at them. In the meanwhile, you've been - metaphorically speaking - in a superposition of all possible states of wearing socks.
Now, in order to explain this phenomenon, do you need faster-than-light travel of your left sock's color to your right sock at any point?
Yes. More specifically you need non-locality, because you expect both feet to have socks of the same color. Now either this is a hidden variable, or the two socks have to coordinate their color at some point in time.
Hidden variable theories have been proven not to be applicable. That leaves some way of coordination. Which requires non-locality.
1) "Hidden variable theories have been proven not to be applicable." This is flat out wrong. Bohmian mechanics is a deterministic hidden-variable theory of quantum mechanics which can do without a probabilistic interpretation of quantum mechanics by giving up locality. The choice is really between giving up locality or determinism. But you don't have to give up both and the Copenhagen interpretation certainly does not give up both.
2) There is no hidden variable in my example. A hidden variable, by definition, is a variable that is inaccessible to measurements. Moreover, my example doesn't depend on the "feet" at all and would work perfectly well without them. That being said, I think what you're actually referring to is that you already choose a certain color when you put your socks on, i.e. the socks' color is not uncertain even if you decide to not look at them until later. True but this is why half of my comment is discussing exactly this point. I admit, though, that I'm being a bit vague there, so here goes another example: Let's say a friend of yours blindfolds you and flips a coin to determine whether to give you a pair of blue socks or a pair of red socks. He subsequently hands you a pair of the respective color and you put them on and decide not to look at them until later. From your point of view, the chances of your left sock being red/blue is 50% each. However, no matter the outcome of your measurement of looking at your left sock, you can be sure that if you look at the color of your right sock afterwards, its color will always be the same as your left sock's one. The two results are correlated. Now, I hope you won't argue that this is still not quantum mechanics because in QM a sock can be multiple colors at the same time whereas classically/in my example they cannot and will always have a certain color, no matter our (lack of) knowledge. This is exactly the point. If I could explain away quantum mechanics through a classical example like this we wouldn't need a non-local or non-deterministic model of our world. But Bell's inequality tells us otherwise. Anyway, what my example does demonstrate is that even for measurements whose outcome is probabilistic, correlations between the results (or "coordination" as you call it) can perfectly well be explained through prior arrangement, i.e. an arrangement at a point in space and time that lies in the past light-cone of both measuring events. So no faster-than-light traveling is needed.
Now, of course this is not the most rigorous argument one can make. For a more rigorous one please refer to the C* algebras approach to quantum mechanics where "locality" can be defined a bit more precisely and regular quantum mechanics indeed turns out to be a theory respecting locality.
If you imagine Im at 0 on the x-axis, and I create an entangled pair, then send one to L and one to -L, then measure both: can't I, at x=0, verify they (don't) match at time L/c rather than 3L/c?
If it were locally probabilistic, then it should accidentally (not) match some of the time between t=L/c and t=3L/c: it cant be both non-deterministic and local if they never match during that time window.
Seperately: fused anyons in electron gases seem to exhibit non-local qunatum numbers. That's why MS wants to use them for quantum computing. Im not sure your sock example is sufficient to explain the stability of topological-regime quantum values against thermal noise and decoherence.
Thanks for your remarks! Unfortunately, I'm pretty busy this weeek, so I just came here to tell you that I will think about your comment a bit and respond as soon as possible!
