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The GHZ theorem contradicts predetermined local outcomes using perfect quantum correlations. For the GHZ state $(|000⟩+|111⟩)/\sqrt{2}$, the products of Pauli operators obeywith certainty, while $X_1{X}_2{X}_3=+1$ with certainty.

Suppose each site has predetermined outcomes $x_i,y_i\in \{-1,+1\}$, independent of which observable is chosen at the other sites. Multiplying the first three constraints cancels every $y_i^2$, forcing $x_1{x}_2{x}_3=-1$, in contradiction with the quantum measurement prediction. The Pauli operators evade that classical multiplication rule because $XY=-YX$ at the same site. The displayed signs use the plus-GHZ state; changing its relative sign changes the individual correlations consistently. ^[ref]