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We provide a new, quantum inspired, proof for the celebrated claim of[LFKN92] that $P^{\#P} \subseteq IP$. The protocol is fundamentally differentfrom the original sum-check protocol of [LFKN92, Sha92], as well as fromvariants of this proof [GKR08, Mei09], though it still possesses the overallstructure of inductively checking consistency between subsequent step. Theprotocol is inspired by [AAV13]. Hopefully, this protocol will be helpful inmaking progress towards two major open problems: resolving the quantum PCPquestion, and verification of quantum computations using a classical BPPverifier. Given the historical importance of the sum-check protocol inclassical computational complexity, we hope that the new protocol will alsofind applications in classical complexity theory.

A Quantum inspired proof of $P^{\#P} \subseteq IP$