My best understanding is that in general, issues related purely to quantum entanglement are entirely unrelated to causality. As noted in the article, even the "spooky action at a distance" that made Einstein so uncomfortable cannot be used to transmit information faster than light. I haven't checked the math, but I assume that the same sort of properties mean that this system couldn't be used to transmit information into the past.
So issues of causality don't really arise: this is just a neat demonstration that the weird correlations implied by quantum mechanics can be separated in a "direction" that hadn't been tested before. (That is, there's definitely time passing in this system and there is definitely entanglement between objects existing at different times, but there's also no real tension between those facts.)
Let me be a little more precise here. What you basically need to know here is that entanglement describes a correlation between measurements, and you cannot know whether that correlation exists until you collect those measurements together. This is the fundamental reason why it does not "transmit information" faster than light in a certain sense. However, that phrasing is misleading because you can use it to do things which would otherwise require transmitting information faster than light.
I like to illustrate this by a game based on "GHZ states" which I call "Betrayal." The idea is that there are 3 people working together, but I'm going to make one of them work at cross-purposes to the other two. If the team can recover gracefully from my mischief with high probability, then they all win a big cash prize.
The game is simple: the three people can prepare however they want in advance, but then they must go into different (relativistically-separated) rooms, look at a screen with words on it, and hit either a button labelled 0 or a button labeled 1. Then I collect these three numbers and sum them up, to get The Sum. So if Alice hits 1, and Bob hits 0, and Carol hits 1, then the Sum is 2.
On the screens in the rooms, I give them a task. Sometimes I do a "control" experiment: I tell all three of them "Make the sum even," and the team wins if it's even. Sometimes I create a traitor: I tell two of them, "make the sum odd", and one of them "make the sum even", and the team wins if it's odd.
Three classical players cannot beat this game 100% of the time, no matter how they prepare in advance. Three quantum players (i.e., three players sharing an entangled Greenberger-Horne–Zeilinger state) can. So if we repeat the game enough times, they can convince me that they can beat the game with higher probability than the classical limit, and thus win the big cash prize.
What makes the quantum players able to beat the limitation of the classical players with multiple trials?
Every time I read an article discussing quantum mechanics, particularly new results in the field, I get more and more of the feeling that we are just missing something. According to the article entanglement can occur on the scale of lightyears but those entanglements cannot be used to transmit information faster than the speed of light.
The linked article on Schrodinger's Hat seems to be violate another rule about observation, but there's always this caveat that prevents it from violating some quantum principal.
It's just that they have access to an operation which classically doesn't exist, because their probabilities are complex numbers rather than real numbers. (Just as importantly, there are known limits to how great their correlation can be; the nice thing about Betrayal is that you can quickly prove that six classical random variables don't work no matter how they're jointly distributed.)
So what is this strange operation? There exist two nice "superposition over all states" quantum states for the three bits held by the three players:
Separately those states are not entangled: that is, +++ is made from the separable (0 + 1)(0 + 1)(0 + 1) while −−− is made from the separable (0 − 1)(0 − 1)(0 − 1). In both "pure" states any bit pattern from 000 to 111 has equal probability. Quantum mechanics now lets these observers have the superposition state:
(+++) + (−−−) = 000 + 011 + 101 + 110
This is an entangled state. In this state you cannot be sure which of these four will occur, but they will each occur with even probability and the sum will be even. So that's the "control" experiment covered. But we could solve the "control" experiment with the 000 state too. What about the "traitor" experiment?
Here's where you need the complex numbers. Each of the "make the sum odd" people maps (+),(−) → (+),i(−). This is called a phase rotation, and you might know i² = -1 in the complex plane. These separate acts shift the global state to:
(+++) + (−−−) → (+++) + i²(−−−) = (+++) − (−−−)
If you work it out you will find:
(+++) − (−−−) = 001 + 010 + 100 + 111
So even though locally nobody can tell what's happened (every single person still has a 50/50 chance of seeing 0 or 1 by themselves), the global sum changes due to this phase rotation. That is what entanglement can get you, large-scale correlations.
As for proving that classical probabilities cannot do this, take six random variables no matter their joint distribution, call them Ao, Ae, Bo, Be, and Co, Ce -- what Alice, Bob, and Carol do when they're told to make the sum odd or even, respectively. The problem asks to make Ao + Bo + Ce ≡ Ao + Be + Co ≡ Ae + Bo + Co ≡ 1 (mod 2) while Ae + Be + Ce ≡ 0 (mod 2). Adding those four equations together gives 2 * (Ao + Bo + Co + Ae + Be + Ce) ≡ 3 (mod 2), but 3 isn't even. So it's not possible to satisfy all four equations all of the time with classical probabilities.
So issues of causality don't really arise: this is just a neat demonstration that the weird correlations implied by quantum mechanics can be separated in a "direction" that hadn't been tested before. (That is, there's definitely time passing in this system and there is definitely entanglement between objects existing at different times, but there's also no real tension between those facts.)