  We provide a generic construction to turn any classical Zero-Knowledge (ZK)
protocol into a composable (quantum) oblivious transfer (OT) protocol, mostly
lifting the round-complexity properties and security guarantees
(plain-model/statistical security/unstructured functions...) of the ZK protocol
to the resulting OT protocol. Such a construction is unlikely to exist
classically as Cryptomania is believed to be different from Minicrypt.
  In particular, by instantiating our construction using Non-Interactive ZK
(NIZK), we provide the first round-optimal (2-message) quantum OT protocol
secure in the random oracle model, and round-optimal extensions to string and
k-out-of-n OT.
  At the heart of our construction lies a new method that allows us to prove
properties on a received quantum state without revealing additional information
on it, even in a non-interactive way, without public-key primitives, and/or
with statistical guarantees when using an appropriate classical ZK protocol. We
can notably prove that a state has been partially measured (with arbitrary
constraints on the set of measured qubits), without revealing any additional
information on this set. This notion can be seen as an analog of ZK to quantum
states, and we expect it to be of independent interest as it extends complexity
theory to quantum languages, as illustrated by the two new complexity classes
we introduce, ZKstatesQIP and ZKstatesQMA.
