Quantum Advantage in Proof Systems Without Entanglement

Abstract

The study of interactive proofs in the quantum setting has yielded profound insights in complexity theory and quantum information. A curious feature of these results is that the advantage, in terms of computational power, of quantum models over their classical counterparts is usually due to entanglement phenomena rather than quantum communication with the verifier. For example, it is known that QIP = IP = PSPACE, and QMIP with unentangled provers is equal to NEXP = MIP; on the other hand, MIP* = RE.
In this work we initiate the general study of (quantum) positional multi-prover interactive proofs ((Q)PMIP), in which provers and verifiers positioned in space communicate freely save for the constraints imposed by the speed of light. We investigate how the class of languages decidable by (Q)PMIPs depends on the arrangement of the verifiers and (honest) provers. In the case of classical PMIPs, we show a dichotomy: if the arrangement satisfies what we call the "min-ball" condition, then the class is NEXP, otherwise it is PSPACE. We then exhibit an arrangement that does not satisfy the min-ball condition for which there is a quantum PMIP for EXP in the no pre-shared entanglement model.
Our construction is based on positional cryptography and MIPs with no-signaling soundness. We introduce a new positional primitive, the positional hardcore bit, which allows a pair of spatially separated players to transmit a random bit to a particular location while guaranteeing that it remains strongly unguessable elsewhere.