Classical verification of quantum computation without cryptographic assumptions
In plain words
A purely classical user can check a quantum computer's answer if one assumes certain codes are hard to break, or if two separated quantum computers are questioned. Whether one quantum computer can be checked by a classical user with no such assumption is open.
Precise statement
Does every language in BQP have an interactive proof with a BPP verifier exchanging only classical messages with a single BQP prover, sound against computationally unbounded provers? Known protocols need a verifier holding a few qubits, two non-communicating entangled provers (Reichardt, Unger and Vazirani 2013), or soundness only against provers who cannot solve learning with errors (Mahadev, arXiv:1804.01082, 2018).
What would settle it
An information-theoretically sound single-prover classical-verifier protocol for BQP, or a complexity-theoretic consequence (such as a class collapse) showing it cannot exist.