  We study the relationship between quantum chaos and pseudorandomness by
developing probes of unitary design. A natural probe of randomness is the
"frame potential," which is minimized by unitary $k$-designs and measures the
$2$-norm distance between the Haar random unitary ensemble and another
ensemble. A natural probe of quantum chaos is out-of-time-order (OTO)
four-point correlation functions. We show that the norm squared of a
generalization of out-of-time-order $2k$-point correlators is proportional to
the $k$th frame potential, providing a quantitative connection between chaos
and pseudorandomness. Additionally, we prove that these $2k$-point correlators
for Pauli operators completely determine the $k$-fold channel of an ensemble of
unitary operators. Finally, we use a counting argument to obtain a lower bound
on the quantum circuit complexity in terms of the frame potential. This
provides a direct link between chaos, complexity, and randomness.
