  We show the following unconditional results on quantum commitments in two
related yet different models:
  1. We revisit the notion of quantum auxiliary-input commitments introduced by
Chailloux, Kerenidis, and Rosgen (Comput. Complex. 2016) where both the
committer and receiver take the same quantum state, which is determined by the
security parameter, as quantum auxiliary inputs. We show that
computationally-hiding and statistically-binding quantum auxiliary-input
commitments exist unconditionally, i.e., without relying on any unproven
assumption, while Chailloux et al. assumed a complexity-theoretic assumption,
${\bf QIP}\not\subseteq{\bf QMA}$. On the other hand, we observe that achieving
both statistical hiding and statistical binding at the same time is impossible
even in the quantum auxiliary-input setting. To the best of our knowledge, this
is the first example of unconditionally proving computational security of any
form of (classical or quantum) commitments for which statistical security is
impossible. As intermediate steps toward our construction, we introduce and
unconditionally construct post-quantum sparse pseudorandom distributions and
quantum auxiliary-input EFI pairs which may be of independent interest.
  2. We introduce a new model which we call the common reference quantum state
(CRQS) model where both the committer and receiver take the same quantum state
that is randomly sampled by an efficient setup algorithm. We unconditionally
prove that there exist statistically hiding and statistically binding
commitments in the CRQS model, circumventing the impossibility in the plain
model.
  We also discuss their applications to zero-knowledge proofs, oblivious
transfers, and multi-party computations.
