  Quantum chaos cannot develop faster than $\lambda \leq 2 \pi/(\hbar \beta)$
for systems in thermal equilibrium [Maldacena, Shenker & Stanford, JHEP
(2016)]. This `MSS bound' on the Lyapunov exponent $\lambda$ is set by the
width of the strip on which the regularized out-of-time-order correlator is
analytic. We show that similar constraints also bound the decay of the spectral
form factor (SFF), that measures spectral correlation and is defined from the
Fourier transform of the two-level correlation function. Specifically, the
inflection exponent $\eta$, that we introduce to characterize the early-time
decay of the SFF, is bounded as $\eta\leq \pi/(2\hbar\beta)$. This bound is
universal and exists outside of the chaotic regime. The results are illustrated
in systems with regular, chaotic, and tunable dynamics, namely the
single-particle harmonic oscillator, the many-particle Calogero-Sutherland
model, an ensemble from random matrix theory, and the quantum kicked top. The
relation of the derived bound with other known bounds, including quantum speed
limits, is discussed.
