Complexity in Krylov basis at extreme scenarios

In recent times, quantum computational complexity has been a major interest to both QI and HEP communities. Complexity in the Krylov basis is one such candidate which has proved its worth in several setups by providing us with meaningful insights. The basic protocol is to construct a consistent Hilbert space to describe the spread of a state (/operator) with time and track the system's evolution by the scaling of the complexity. I will discuss how and why this notion appears to be such a useful probe into the system dynamics by studying some models ranging from the simplest quantum mechanical setups of unitary circuits to the extreme quantum gravity setups of highly excited string scattering.