Self-testing is a powerful and widely used technique in the study of non-local games. Informally, a game self-tests a target entangled state if any state that forms part of an optimal quantum strategy can be mapped to the target state via a local isometry acting on each player's space.

While self-testing allows us to characterize the entanglement shared between non-communicating players, the requirement that the isometries be local limits its applicability when the goal is to certify that a specific player locally holds a desired quantum state—such as witness to a QMA problem.

In this talk, I will present how one can make sense of a form of self-testing without the locality constraint on the isometry. More precisely, we introduce and formalize the notion of classical proofs of quantum knowledge (cPoQ) for non-local games, and show how this framework can be used to certify that the players' joint strategy involves possession of a specific quantum state—namely, the ground state of a local Hamiltonian.

This is based on joint works with: Anne Broadbent, Alex Grilo, Nagisa Hara, and Yuming Zhao.