  We explore spread and spectral complexity in quantum systems that exhibit a
transition from integrability to chaos, namely the mixed-field Ising model and
the next-to-nearest-neighbor deformation of the Heisenberg XXZ spin chain. We
corroborate the observation that the presence of a peak in spread complexity
before its saturation, is a characteristic feature in chaotic systems. We find
that, in general, the saturation value of spread complexity post-peak depends
not only on the spectral statistics of the Hamiltonian, but also on the
specific state. However, there appears to be a maximal universal bound
determined by the symmetries and dimension of the Hamiltonian, which is
realized by the thermofield double state (TFD) at infinite temperature. We also
find that the time scales at which the spread complexity and spectral form
factor change their behaviour agree with each other and are independent of the
chaotic properties of the systems. In the case of spectral complexity, we
identify that the key factor determining its saturation value and timescale in
chaotic systems is given by minimum energy difference in the theory's spectrum.
This explains observations made in the literature regarding its earlier
saturation in chaotic systems compared to their integrable counterparts. We
conclude by discussing the properties of the TFD which, we conjecture, make it
suitable for probing signatures of chaos in quantum many-body systems.
