  Chaos, in quantum systems, can be diagnosed by certain out-of-time-order
correlators (OTOCs) that obey the chaos bound of Maldacena, Shenker, and
Stanford (MSS). We begin by deriving a dispersion relation for this class of
OTOCs, implying that they must satisfy many more constraints beyond the MSS
bound. Motivated by this observation, we perform a systematic analysis
obtaining an infinite set of constraints on the OTOC. This infinite set
includes the MSS bound as the leading constraint. In addition, it also contains
subleading bounds that are highly constraining, especially when the MSS bound
is saturated by the leading term. These new bounds, among other things, imply
that the MSS bound cannot be exactly saturated over any duration of time,
however short. Furthermore, we derive a sharp bound on the Lyapunov exponent
$\lambda_2 \le \frac{6\pi}{\beta}$ of the subleading correction to maximal
chaos.
