We prove that Kilian’s four-message succinct argument system is post-quantum
secure in the standard model when instantiated with any probabilistically
checkable proof and any collapsing hash function (which in turn exist based on
the post-quantum hardness of Learning with Errors). This yields the first
post-quantum succinct argument system from any falsifiable assumption.

At the heart of our proof is a new quantum rewinding procedure that enables a
reduction to repeatedly query a quantum adversary for accepting transcripts as
many times as desired. Prior techniques were limited to a constant number of
accepting transcripts.

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