Quantitative Quantum Soundness for Bipartite Compiled Bell Games via the Sequential NPA Hierarchy

Compiling Bell games under cryptographic assumptions replaces the need for physical separation, allowing nonlocality to be probed with a single untrusted device. While Kalai et al. (STOC'23) showed that this compilation preserves quantum advantages, its quantitative quantum soundness has remained an open problem. We address this gap with two primary contributions. First, we establish the first quantitative quantum soundness bounds for all bipartite compiled Bell games via a newly formalized convergent sequential Navascués-Pironio-Acín (NPA) hierarchy: any polynomial-time prover’s score is controlled by a finite-level hierarchy value, and finite-level convergence gives a negligible gap to the commuting quantum value, or to the tensor-product quantum value under flat optimality. Second, we develop the sequential NPA hierarchy as a computational tool of independent interest, establishing its convergence and a stopping criterion based on flatness, and identifying its conic dual as a sparse SOS hierarchy.