The classical idea is more like as follows: you have a guy Charles who makes two letters each with a card in that has written on it either number 0 or 1, and gives one letter to Alice and another to Bob. Now, "entanglement" here is that if Charles writes 0 in Alice's letter he writes the number 1 in Bob's, and vice versa. Alice and Bob are also aware of this. This means that even at a distance, if Alice opens her letters and sees a 0, she knows that Bob's has 1 in.

For a long time, Einstein and co. through the "EPR paradox" (Einstein-Podolsky-Rosen) thought that QM would be something like this, but we just didn't know how to access the actual letter "in transit" yet, which they called an "element of reality". They were wrong.

It turns out that Bell's theorem basically says that in this situation the correlation between the two letters would be less strong than quantum theory predicts for entangled states, so this cannot be the full explanation.

Bell's theorem says you either need to give up the idea that there is no "spooky action", or give up determinism (no true "element of reality" that really holds what's going on), or you can retain both if you let go of another concept called statistical independence.

> Have an array of 8 particles at both locations. They represent measured/unmeasured rather than heads/tails. You got yourself a one-way single-use FTL byte.

You can't have a quantum state where something is the superposition of measured and not measured, this violates a fundamental postulate of quantum mechanics: that measurement "chooses" one of the states in the measurement basis. I think you touch on that later, in that you can't tell via entanglement if a measurement took place elsewhere, which is partly why we can't experimentally demonstrate a non-local theory yet.

Basically, it isn't that some signal is transmitted, it's that if they are entangled the states of each system cannot be independently known. That is, like in the letters example I gave: it can be either 0 for Alice and 1 for Bob, or 0 for Alice and 1 for Bob. Now, the reason why information is not transmitted is the same: Bob didn't know what was in his letter before he measured it, so it's randomly symmetric between 0 and 1, there is no information transferred. It turns out that this kind of symmetry between the two states tells you nothing that can be used to transmit information.