  We present two protocols for classical verification of quantum depth. Our
protocols allow a purely classical verifier to distinguish devices with
different quantum circuit depths even in the presence of classical computation.
We show that a device with quantum circuit depth at most d will be rejected by
the verifier even if the prover applies additional polynomial-time classical
computation to cheat. On the other hand, the verifier accepts a device which
has quantum circuit depth d' for some d'>d. In our first protocol, we introduce
an additional untrusted quantum machine which shares entanglements with the
target machine. Applying a robust self-test, our first protocol certifies the
depth of the target machine with information theoretic security and nearly
optimal separation. The protocol relies on the oracle separation problem for
quantum depth by Chia, Chung and Lai [STOC 2020] and a transformation from an
oracle separation problem to a two-player non-local game. Our second protocol
certifies the quantum depth of a single device based on quantum hardness of
learning with errors. The protocol relies on the noisy trapdoor claw-free
function family and the idea of pointer chasing to force the prover to keep
quantum coherence until all preceding message exchanges are completed. To our
knowledge, we give the first constructions for distinguishing hybrid
quantum-classical computers with different circuit depths in unrelativized
models.
