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But here you have two independent coins. My question is, how is entanglement fundamentally different than having two sides of the SAME coin but not looking until later?

Let's forget how this looks like two correlated coins and focus on your question.

Yes, if Alice and Bob just measure the spin of their corresponding electron, indeed it just looks like half of a precut coin. But if they want to win the game they will do something more complicated. (side note: To use the typical physicist terminology, this precut coin is a hidden variable - something set inside of the electron even before its measurement.)

But Alice and Bob can do something more interesting, something that permits them to win the game we described more often than 75% of the time. This involves Alice measuring the spin of the electron in a projection different than the one in which Bob is measuring (this is part of the "specific protocol" that I mentioned in the previous post). So while Alice is checking whether the result is h=Heads or t=Tails, Bob is checking whether the result is h+t or h-t. Now the two electrons seize to behave like the two parts of the same coin (and this is what permits them to win the game more often than 75% of the time). Explaining how exactly they use this protocol would take too much space, but you can look it up in the link to the book I provided.

P.S. However, as explained in the previous post, this does not permit them to send messages to each other.

P.P.S. Again, Section 4 of the linked page explains this in details!

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