  In this paper, we studied a set of generalised Krylov complexity for operator
growth. We demonstrate their universal features at both initial times and long
times using half-analytical technique as well as numerical results. In
particular, by using the logarithmic relation to the Krylov entropy, we
establish an inequality (\ref{master}) between the variance of the K-complexity
and the generalised notions which holds in the long time limit. Extending the
result to finite (but long) times, we show that for fast scramblers, the
K-complexity constrains the growth of generalised complexity more stringently
than the dispersion bound. However, for slow scramblers, the growth rate of
K-complexity is tighter bounded by the generalised complexity in the other way
around. Our results enlarge the zoo of Krylov quantities and may shed new light
on the future research in this field.
