= GHZ game
{c}
A three-party <nonlocal game> with binary questions of even <Hamming weight>. Its winning output parity is half the question weight modulo two. For the four possible input triples, a hypothetical deterministic classical response would give four parity equations whose left sides cancel under <exclusive or>, while their right sides have odd total parity. Thus no classical strategy succeeds on every allowed triple, even with pre-shared randomness.
On the <GHZ state>, measure the <Pauli X gate> as an observable for question zero and the <Pauli Y gate> for question one, and encode the measured <eigenvalue> as $(-1)^x$. The joint observables XXX, XYY, YXY and YYX have respective GHZ <eigenvalues> $+1,-1,-1,-1$. Their products of local outcomes enforce the required parity with certainty. The strategy uses pre-shared <quantum entanglement> and local measurements during the game, without communication. It realizes the <GHZ theorem> as an operational task. If the four questions are uniform, the best classical success is $3/4$: the parity contradiction bounds it above, and the deterministic rule $x=a$, $y=z=0$ wins three triples and loses only 011. The uniformity assumption is necessary for that numerical score, not for the impossibility of worst-case certainty.
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