So not having yet read through OP I am not terribly surprised that this is true and I can kind of give a quick sketch in terms of a QM game that I want everyone to know, called Betrayal.
The idea is that it's a collaborative game for three people, you are trying to work together to beat the rules of the game. Meanwhile the rules are trying to set you up so that one of the people betrays the other two. In 3 relativistically separated rooms (so they can’t communicate) they go, where they find a screen and buttons labeled 0 and 1. The screen displays a prompt, each teammate presses exactly one of the buttons once before time runs out, then the three numbers pressed get summed together into a number.
25% of the time we run a “control round,” everyone gets a prompt to make the sum of their numbers even, and they win if the sum is even. The easiest way is if everyone hits 0, 0+0+0 is even. But a team can also answer 0+1+1 or so and win. Otherwise we randomly choose one to be the traitor and send them the control prompt, to make the sum even. But we send the other two the prompt to make the sum odd! In this case the team will only win if their joint sum is odd.
Long story short, classical players of this game have a success probability bounded from above by 75%. This is the Bell inequality. But quantum capable players can walk in with a GHZ state,
|+++> + |–––>,
which only collapses to even sums. If they have to do a control round they will all just measure this in the computational basis.
The more interesting thing, where i really matters, comes during the traitor rounds. Here you want to perform the phase rotation gate in the Hadamard basis,
|+> → |+>,
|–> → i |–>,
And any two of them can thereby switch the state to
|+++> – |–––>,
a state which only has odd configurations. Quantum players can win 100% of the time. Over multiple independent trials you should be able to observe the inequality violations even if quantum coherence were to limit your success probability to 90%.
I suspect that the inequality here is something similar, quantum mechanics but you can only form real-coefficient superpositions, and therefore you cannot take the square root of a unitary transformation just by doing it for half the time, per Schrödinger.
ELI5: Why is a typical phone call today less intelligible than a 8kHz 8-bit μ-law with ADPCM from the '90s did?
[edit]
s/sound worse/less intelligible/