Commitments are equivalent to statistically-verifiable one-way state generators

One-way state generators (OWSG) are natural quantum analogs to classical one-way functions. We consider statistically-verifiable OWSGs (sv-OWSG), which are potentially weaker objects than OWSGs. We show that O(n/log(n))-copy sv-OWSGs (n represents the input length) are equivalent to poly(n)-copy sv-OWSGs and to quantum commitments. Since known results show that o(n/log(n))-copy OWSGs cannot imply commitments, this shows that O(n/log(n))-copy sv-OWSGs are the weakest OWSGs from which we can get commitments (and hence much of quantum cryptography). Our construction follows along the lines of Hastad, Impagliazzo, Levin and Luby, who obtained classical pseudorandom generators (PRG) from classical one-way functions (OWF), however with crucial modifications. Our construction, when applied to the classical case, provides an alternative to the classical construction to obtain a classical mildly non-uniform PRG from any classical OWF. Since we do not argue conditioned on the output $f(x)$, our construction and analysis is arguably simpler and may be of independent interest. For converting a mildly non-uniform PRG to a uniform PRG, we can use the classical construction.