  In this work, we investigate spectral complexity and Krylov complexity in
quantum billiard systems at finite temperature. We study both circle and
stadium billiards as paradigmatic examples of integrable and non-integrable
quantum-mechanical systems, respectively. We show that the saturation value and
time scale of spectral complexity may be used to probe the non-integrability of
the system since we find that when computed for the circle billiard, it
saturates at a later time scale compared to the stadium billiards. This
observation is verified for different temperatures. Furthermore, we study the
Krylov complexity of the position operator and its associated Lanczos
coefficients at finite temperature using the Wightman inner product. We find
that the growth rate of the Lanczos coefficients saturates the conjectured
universal bound at low temperatures. Additionally, we also find that even a
subset of the Lanczos coefficients can potentially serve as an indicator of
integrability, as they demonstrate erratic behavior specifically in the circle
billiard case, in contrast to the stadium billiard. Finally, we also study
Krylov entropy and verify its early-time logarithmic relation with Krylov
complexity in both types of billiard systems.
