Perfect play with spatial quantum strategies is Σ⁰₂-complete

When does a finite nonlocal game admit perfect spatial quantum play? We prove that this decision problem is complete for the second existential level of the arithmetical hierarchy. The same classification holds when a perfect strategy must have finite marginal entropy. Hardness persists among games whose finite-dimensional value is one and which have no finite-dimensional perfect strategy. A finite clock-phase order supplies the existential witness: normality forces a finite order on a nonzero sector, while the strategy chooses the order. For the upper bound, we normalize the largest Schmidt coefficient and use compact operator sandwiches. Fixing a lower bound on the largest Schmidt probability gives a co-recursively enumerable stratum. No computable bound selects a sufficient stratum for every game. Exact attainment of a rational Bell bound, of an unspecified supremum, and of a specified scalar value is also Sigma^0_2-complete. The hardness proof extends the modular-shift and rational-row toolkit of our earlier entropy-game paper to four calibrated bridges and a deterministic computation on rooted finite squares.