Why must entangled particles communicate their spin instantaneously?

Do they explicitly say "communicate", or are you paraphrasing? Because communicate's wrong. At least, when the spin measurement events are separated by a space-like interval (which are the situations where entanglement's mysterious), communication's >>impossible<<. So in those situations, entangled particles simply aren't communicating.

Let me give you a non-mysterious classical example. Suppose you hide two balls in two identical-looking boxes, a red ball in one and a blue ball in the other. Now, I hasten to immediately say that's >>not<< like entanglement -- the ball colors are predetermined (I'll give you a non-predetermined classical-like example below). Anyway, suppose you now separate your two boxes by several miles (or light-years if you like). If somebody opens one box and sees a red ball, then we know somebody opening the other box will see blue. Right? But no "communication" is necessary. Right?

Well, that's exactly what happens with entangled particles. No mystery, except for the following, which is where that promised non-predetermined example comes in, as follows. Instead of two balls, suppose each box contains a coin that you flipped in the air, and the coins are still spinning. They'll eventually land heads-or-tails, but they don't land until their respective box is opened. The mystery is that if they're "entangled", then if one lands heads, the other lands tails, and vice versa. So you'd infer that there must somehow be some "communicating" going on.

However, by analogy with our spin-entangled particles, we know for sure there's no communication. And that's the mystery -- we can't intuitively reconcile our classical thinking, as illustrated by our balls/coins examples, with that nonclassical reality. To summarize the thinking illustrated by those classical examples, recall that the balls don't communicate but there's no mystery about the correlation between their outcomes. The coins also don't communicate, but now any observed correlation is unexpected/mysterious. Ditto the spin-entangled particles. But "communicating" isn't the answer (nobody yet has an intuitively satisfying answer).

Edit since the comment below got several up-votes, let me expand that remark a bit as follows...

The observed correlation between measurements of entangled particles, whenever the measurement events are separated by a space-like interval, cannot be explained by cause-and-effect "communication". Call the two measurement events $m_1$ and $m_2$. Then one observer may see $m_1$ occurring before $m_2$ while a second observer (in a different inertial frame) may see $m_2$ occurring before $m_1$. So there's no unambiguous way to say which came first, the   $m_1=\mbox{chicken}$   or the   $m_2=\mbox{egg}$   (sorry, I couldn't resist:). So you cannot say one "caused" the other, simply because you can't even say one occurred before the other.

The only thing you can unambiguously say is that the single preparation event that prepared both particles in a single entangled state occurred before both subsequent measurements (it's in both their past light cones). And it's that preparation event which somehow "caused" the subsequent measurement correlation. That is, it didn't cause each individual measurement outcome, per se, because they're ultimately random, but it did somehow cause the correlation between them.

And while you're probably thinking, "Huh?...how do you cause a correlation without causing the correlated events?", that's likely a problem with our intuition developed over a lifetime observing classical phenomena.

A very good question because "instantaneous" is not really meaningful within the framework of special relativity. "Instantaneous" would imply the observation of the two particle states at the same time, but that will depend on our reference frame. See https://en.wikipedia.org/wiki/Spacetime#Relativity_of_simultaneity.

What they really mean in these videos, and what experiments reveal, is that the states of the two particles are correlated even when the two measurements take place outside of each other's light cones. In other words we see correlation even when there is a spacelike separation between the two measurements. Current theories do not allow information travel between the two spacelike separated measurements. (BTW with spacelike separation we can always find a reference frame whose observer will see the two measurements really happen at the same time.)

You might be interested in this article about a specific 2013 experiment. The events are laid out in this diagram:

There is no (known) way that the result of measurement A could have influenced the outcome of measurement B or vice versa. But "communicating instantaneously" is still grossly misleading.

Here is an experimentalist's answer:

Quantum mechanics is the underlying framework of nature from which the classical framework emerges. It is a mathematical model which depends on postulates, like extra axioms, to pick from all the possible solutions of the quantum mechanical wave equations , those that fit the data. Up to now there has been no falsification in these calculations. i.e. they describe existing data and predict new situations successfully.

The QM model predicts probability densities, i.e. the mathematical solutions, the wavefunctions which are defined by the boundary conditions of the problem, when complex conjugate squared, give the probability density distribution for an interaction or a decay or.... Experimentally probability densities are built up event per event, and then the distributions are checked against the predictions of the theory.

The wavefunction is not a measurable quantity, it is only the accumulation of many events that can validate the wavefunction.

Wavefunctions cover all of the available phase space in space and time, and the conservation laws hold for individual events. For example the wavefunction for scattering two electrons contains all the probabilities for the scattering of an electron on another electron and their spins summing to zero, all the spin information is contained in the wavefunction. When a single event is measured, i.e. one electron is measured to have spin 1/2 , conservation laws will tell us that the other has spin -1/2, immediately. There is no transfer of information.

Things get more complicated for solutions of relativistic differential equations, because there may be solutions/wavefunctions which may have a mathematical description outside the light cone. Since we know that faster than light communication is experimentally excluded, it means that this particular solution cannot be used to describe a physically meaningful situation. See my answer to a relevant question here.

Entanglement is a shorthand way of saying there exists a complete quantum mechanical solution for theses boundary conditions and this system. If the solution is valid, it contains by construction all the information, nothing is communicated, it is just described/modeled.