  Krylov complexity measures the spread of the wavefunction in the Krylov
basis, which is constructed using the Hamiltonian and an initial state. We
investigate the evolution of the maximally entangled state in the Krylov basis
for both chaotic and non-chaotic systems. For this purpose, we derive an
Ehrenfest theorem for the Krylov complexity, which reveals its close relation
to the spectrum. Our findings suggest that neither the linear growth nor the
saturation of Krylov complexity is necessarily associated with chaos. However,
for chaotic systems, we observe a universal rise-slope-ramp-plateau behavior in
the transition probability from the initial state to one of the Krylov basis
states. Moreover, a long ramp in the transition probability is a signal for
spectral rigidity, characterizing quantum chaos. Also, this ramp is directly
responsible for the late-time peak of Krylov complexity observed in the
literature. On the other hand, for non-chaotic systems, this long ramp is
absent. Therefore, our results help to clarify which features of the wave
function time evolution in Krylov space characterize chaos. We exemplify this
by considering the Sachdev-Ye-Kitaev model with two-body or four-body
interactions.
