  We study the relationship between notions of pseudorandomness in the quantum
and classical worlds. Pseudorandom quantum state generator (PRSG), a
pseudorandomness notion in the quantum world, is an efficient circuit that
produces states that are computationally indistinguishable from Haar random
states. PRSGs have found applications in quantum gravity, quantum machine
learning, quantum complexity theory, and quantum cryptography. Pseudorandom
generators, on the other hand, a pseudorandomness notion in the classical
world, is ubiquitous to theoretical computer science. While some separation
results were known between PRSGs, for some parameter regimes, and PRGs, their
relationship has not been completely understood.
  In this work, we show that a natural variant of pseudorandom generators
called quantum pseudorandom generators (QPRGs) can be based on the existence of
logarithmic output length PRSGs. Our result along with the previous separations
gives a better picture regarding the relationship between the two notions. We
also study the relationship between other notions, namely, pseudorandom
function-like state generators and pseudorandom functions. We provide evidence
that QPRGs can be as useful as PRGs by providing cryptographic applications of
QPRGs such as commitments and encryption schemes.
  Our primary technical contribution is a method for pseudodeterministically
extracting uniformly random strings from Haar-random states.
