I'm a theoretical physicist, I've thought about entanglement for decades. It's magic: if I gave you infinite (classical) engineering powers, you could not achieve what entanglement can do.<p>Unfortunately the typical CHSH inequality is not beginner-friendly because there are a few conceptual steps chained together before one can understand the whole idea. However one can rewrite it in terms of a game. Here's the GHZ game, which is about 3 particle entanglement and the classical/quantum separation is much more clear. Here it is.<p>There is a team of three players and a referee. The players can agree on strategies all they want, but during the game they cannot communicate. The referee sends a challenge bit to each of the players, so the referee emits the three bits (r,s,t). They all have to reply with a bit, so the referee receives the three bits (x,y,z). The players win the game if r AND s AND t = x XOR y XOR z. The best they can hope to do if they have access to classical resources is to win 75% of the time. BUT if they share a certain 3-qubit state, they are allowed to make measurements of their qubit and base their answer on the measurement outcome. Well, there is a quantum strategy that lets them win 100% of the time.<p>There is no collection of 6 bits (challenge and reply) that makes you win 100% of the time. So the utter magic of entanglement is that the VALUE of the outcomes of the measurements cannot be thought of as pre-existing the measurement. Because if it did there would be no way to win 100% of the time.