  We show the following hold, unconditionally unless otherwise stated, relative
to a random oracle with probability 1:
  - There are NP search problems solvable by BQP machines but not BPP machines.
  - There exist functions that are one-way, and even collision resistant,
against classical adversaries but are easily inverted quantumly. Similar
separations hold for digital signatures and CPA-secure public key encryption
(the latter requiring the assumption of a classically CPA-secure encryption
scheme). Interestingly, the separation does not necessarily extend to the case
of other cryptographic objects such as PRGs.
  - There are unconditional publicly verifiable proofs of quantumness with the
minimal rounds of interaction: for uniform adversaries, the proofs are
non-interactive, whereas for non-uniform adversaries the proofs are two message
public coin.
  - Our results do not appear to contradict the Aaronson-Ambanis conjecture.
Assuming this conjecture, there exist publicly verifiable certifiable
randomness, again with the minimal rounds of interaction.
  By replacing the random oracle with a concrete cryptographic hash function
such as SHA2, we obtain plausible Minicrypt instantiations of the above
results. Previous analogous results all required substantial structure, either
in terms of highly structured oracles and/or algebraic assumptions in
Cryptomania and beyond.
