  Pseudorandom quantum states (PRS) are efficiently constructible states that
are computationally indistinguishable from being Haar-random, and have recently
found cryptographic applications. We explore new definitions, new properties
and applications of pseudorandom states, and present the following
contributions:
  1. New Definitions: We study variants of pseudorandom function-like state
(PRFS) generators, introduced by Ananth, Qian, and Yuen (CRYPTO'22), where the
pseudorandomness property holds even when the generator can be queried
adaptively or in superposition. We show feasibility of these variants assuming
the existence of post-quantum one-way functions.
  2. Classical Communication: We show that PRS generators with logarithmic
output length imply commitment and encryption schemes with classical
communication. Previous constructions of such schemes from PRS generators
required quantum communication.
  3. Simplified Proof: We give a simpler proof of the Brakerski--Shmueli
(TCC'19) result that polynomially-many copies of uniform superposition states
with random binary phases are indistinguishable from Haar-random states.
  4. Necessity of Computational Assumptions: We also show that a secure PRS
with output length logarithmic, or larger, in the key length necessarily
requires computational assumptions.
