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.
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.
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.
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.
One outcome of this is that faster-than-light communication is definitely a thing, but we can't use it.
An analogy is that there are two telegraph operators in different cities. You write a message on a piece of paper and hand it to the operator in St. Joseph, Missouri. He transcribes it into Morse code and sends it to another operator in San Francisco, who transcribes it back into English for the recipient there.
EPR pairs are like those telegraph operators, except they can communicate with each other instantly across the entire universe.
Wow!
Unfortunately they also have a few caveats:
1. The sending operator completely ignores the message you wrote down and sends a random message instead. You can control when the message is sent and you can see what message was sent, but you cannot control the contents of the message.
2. The recipient can see what message was received on the receiving end, and they can prove it is causally correlated with the sent message by using out-of-band (i.e. ordinary sublight) communication.
3. Once a [random] 1-bit message has been sent, both operators cut the telegraph wires and they can never communicate with each other again.
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.
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.
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.