  The spectral form factor is a powerful probe of quantum chaos that diagnoses
the statistics of energy levels, but is blind to other features of a theory
such as matrix elements of operators or OPE coefficients in conformal field
theories. In this paper, we introduce generalized spectral form factors: new
probes of quantum chaos sensitive to the dynamical data of a theory. These
quantities can be studied using an effective theory of quantum chaos. We focus
our attention on a particular combination of heavy-heavy-heavy OPE coefficients
that generalizes the genus-2 partition function of two-dimensional CFTs, for
which we define a spectral form factor. We probe heavy-heavy-heavy OPE
coefficients and find statistical correlations that agree with the OPE
Randomness Hypothesis: these coefficients have a random matrix component in the
ergodic regime. The EFT of quantum chaos predicts that the genus-2 spectral
form factor displays a ramp and a plateau. Our results suggest that this is a
common property of generalized spectral form factors.
