  How could quantum cryptography help us achieve what are not achievable in
classical cryptography? In this work we consider the following problem, which
we call succinct RSPV for classical functions (SRC). Suppose $f$ is a function
described by a polynomial time classical Turing machine, which is public; the
client would like to sample a random $x$ as the function input and use a
protocol to send $f(x)$ to the server. What's more, (1) when the server is
malicious, what it knows in the passing space should be no more than $f(x)$;
(2) the communication should be succinct (that is, independent to the running
time of evaluating $f$). Solving this problem in classical cryptography seems
to require strong cryptographic assumptions.
  We show that, perhaps surprisingly, it's possible to solve this problem with
quantum techniques under much weaker assumptions. By allowing for quantum
communication and computations, we give a protocol for this problem assuming
only collapsing hash functions [Unr16]. Our work conveys an interesting message
that quantum cryptography could outperform classical cryptography in a new type
of problems, that is, to reduce communications in meaningful primitives without
using heavy classical cryptographic assumptions.
