Unbounded degree overhead for Alice-conditioned quantum Bell certificates

Abstract

Requiring each sum-of-squares term to involve only one of Alice's measurement questions can impose an unbounded certification cost. In the simplest Bell scenario, we prove that no finite level of the Alice-conditioned NPA hierarchy contains all standard level-two Bell certificates. An explicit family of truncated positive functionals on the infinite dihedral group exceeds the tilted-CHSH quantum bound at every prescribed finite level, while a standard degree-two certificate is exact. A Fejer-weighted trace reduces positivity to a rank-one subtraction from a moving-average Gram matrix. The required conditioned level grows at least as $(2-α)^{-1/2}$ near the endpoint tilt. This bounds the degree of exact nice-SOS inputs to compiled-game soundness proofs. Consequently, no finite conditioned level certifies the entire optimal CHSH randomness tradeoff against quantum side information, although standard level two does. Away from the endpoint, we prove a sharp one-level cost throughout $α\in[13/10,3/2]$, using optimal-strategy kernels and exact Bernstein matrix positivity to certify a continuous interval. The results separate ordinary SOS degree from the resources keyed by single-question certificate structure.