  Krylov complexity has recently emerged as a new paradigm to characterize
quantum chaos in many-body systems. However, which features of Krylov
complexity are prerogative of quantum chaotic systems and how they relate to
more standard probes, such as spectral statistics or out-of-time-order
correlators (OTOCs), remain open questions. Recent insights have revealed that
in quantum chaotic systems Krylov state complexity exhibits a distinct peak
during time evolution before settling into a well-understood late-time plateau.
In this work, we propose that this Krylov complexity peak (KCP) is a hallmark
of quantum chaotic systems and suggest that its height could serve as an `order
parameter' for quantum chaos. We demonstrate that the KCP effectively
identifies chaotic-integrable transitions in the mass-deformed
Sachdev-Ye-Kitaev model at both infinite and finite temperature, aligning with
results from spectral statistics and OTOCs. Our findings offer a new diagnostic
tool for quantum chaos that is operator-independent, potentially leading to
more `universal' insights and a deeper understanding of general properties in
quantum chaotic systems.
