  Quantum complexity is a measure of the minimal number of elementary
operations required to approximately prepare a given state or unitary channel.
Recently, this concept has found applications beyond quantum computing -- in
studying the dynamics of quantum many-body systems and the long-time properties
of AdS black holes. In this context Brown and Susskind \cite{BrownSusskind17}
conjectured that the complexity of a chaotic quantum system grows linearly in
time up to times exponential in the system size, saturating at a maximal value,
and remaining maximally complex until undergoing recurrences at
doubly-exponential times. In this work we prove the saturation and recurrence
of complexity in two models of chaotic time evolutions based on (i) random
local quantum circuits and (ii) stochastic local Hamiltonian evolution. Our
results advance an understanding of the long-time behaviour of chaotic quantum
systems and could shed light on the physics of black hole interiors. From a
technical perspective our results are based on establishing new quantitative
connections between the Haar measure and high-degree approximate designs, as
well as the fact that random quantum circuits of sufficiently high depth
converge to approximate designs.
