Sum-of-squares (SOS) proofs are a powerful algebraic tool for analyzing quantum non-local games, enabling bounds on game values and establishing self-testing properties essential for device-independent quantum protocols. A significant practical limitation of these self-tests is the requirement of strictly non-communicating provers. To address this, Kalai et al.\~(STOC '23) introduced a cryptographic compilation method, transforming two-party non-local scenarios into interactions between a classical verifier and a single computationally bounded prover.

This talk reviews recent progress in adapting SOS techniques to the compiled setting. We will highlight how SOS arguments provide bounds and certify self-testing properties for compiled games, discuss current limitations, and outline key open problems.

This talk is based on several joint works with David Cui, Giulio Malavolta, Anand Natarajan, Connor Paddock, Simon Schmidt, William Slofstra, Michael Walter, Lewis Wooltorton, Tina Zhang, and Yuming Zhao..