Commitments are Equivalent to Statistically-Verifiable One-Way State Generators

One-way state generators (OWSG) [1] 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 $O\left(\frac{n}{\log (n)}\right)$ -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 $o\left(\frac{n}{\log (n)}\right)$ -copy OWSGs cannot imply commitments [2], this shows that $O\left(\frac{n}{\log(n)}\right)$ -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 [3], 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 construction provided by [3] 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 same construction as [3].