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.