  Quantum chaotic systems are conjectured to display a spectrum whose
fine-grained features (gaps and correlations) are well described by Random
Matrix Theory (RMT). We propose and develop a complementary version of this
conjecture: quantum chaotic systems display a Lanczos spectrum whose local
means and covariances are well described by RMT. To support this proposal, we
first demonstrate its validity in examples of chaotic and integrable systems.
We then show that for Haar-random initial states in RMTs the mean and
covariance of the Lanczos spectrum suffices to produce the full long time
behavior of general survival probabilities including the spectral form factor,
as well as the spread complexity. In addition, for initial states with
continuous overlap with energy eigenstates, we analytically find the long time
averages of the probabilities of Krylov basis elements in terms of the mean
Lanczos spectrum. This analysis suggests a notion of eigenstate complexity, the
statistics of which differentiate integrable systems and classes of quantum
chaos. Finally, we clarify the relation between spread complexity and the
universality classes of RMT by exploring various values of the Dyson index and
Poisson distributed spectra.
