  We demonstrate quantum advantage with several basic assumptions, specifically
based on only the existence of OWFs. We introduce inefficient-verifier proofs
of quantumness (IV-PoQ), and construct it from classical bit commitments.
IV-PoQ is an interactive protocol between a verifier and a quantum prover
consisting of two phases. In the first phase, the verifier is probabilistic
polynomial-time, and it interacts with the prover. In the second phase, the
verifier becomes inefficient, and makes its decision based on the transcript of
the first phase. If the prover is honest, the inefficient verifier accepts with
high probability, but any classical malicious prover only has a small
probability of being accepted by the inefficient verifier. Our construction
demonstrates the following results: (1)If one-way functions exist, then IV-PoQ
exist. (2)If distributional collision-resistant hash functions exist (which
exist if hard-on-average problems in $\mathbf{SZK}$ exist), then constant-round
IV-PoQ exist. We also demonstrate quantum advantage based on worst-case-hard
assumptions. We define auxiliary-input IV-PoQ (AI-IV-PoQ) that only require
that for any malicious prover, there exist infinitely many auxiliary inputs
under which the prover cannot cheat. We construct AI-IV-PoQ from an
auxiliary-input version of commitments in a similar way, showing that (1)If
auxiliary-input one-way functions exist (which exist if
$\mathbf{CZK}\not\subseteq\mathbf{BPP}$), then AI-IV-PoQ exist. (2)If
auxiliary-input collision-resistant hash functions exist (which is equivalent
to $\mathbf{PWPP}\nsubseteq \mathbf{FBPP}$) or $\mathbf{SZK}\nsubseteq
\mathbf{BPP}$, then constant-round AI-IV-PoQ exist.
