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

Abstract: 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 every bipartite compiled Bell game whose optimal quantum strategy is finite-dimensional: any polynomial-time prover's score in the compiled game is negligibly close to the game's ideal quantum value. More generally, for all bipartite games we show that the compiled score cannot significantly exceed the bounds given by a newly formalized sequential Navascu\'es-Pironio-Ac\'in (NPA) hierarchy. Second, we provide a full characterization of this sequential NPA hierarchy, establishing it as a robust numerical tool that is of independent interest. Finally, for games without finite-dimensional optimal strategies, we explore the necessity of NPA approximation error for quantitatively bounding their compiled scores, linking these considerations to the complexity conjecture $\mathrm{MIP}^{\mathrm{co}}=\mathrm{coRE}$ and open challenges such as quantum homomorphic encryption correctness for "weakly commuting" quantum registers.