Universal Time Evolution of Holographic and Quantum Complexity

Holographic complexity, as the bulk dual of quantum complexity, encodes the geometric structure of black hole interiors. Motivated by the complexity = anything proposal, we introduce the spectral representation for generating functions associated with codimension-one and codimension-zero holographic complexity measures. These generating functions exhibit a universal slope-ramp-plateau structure analogous to the spectral form factor in chaotic quantum systems. In such systems, quantum complexity evolves universally, displaying long-time linear growth followed by saturation at late times. By employing the generating function formalism, we show that this universal behavior has two origins: a particular pole structure of the matrix elements of the generating functions in the energy eigenbasis and random matrix universality in spectral statistics. Using the residue theorem, we prove that the existence of this pole structure is a necessary and sufficient condition for the linear growth of holographic complexity measures. Furthermore, we show that the late-time saturation plateau arises directly from the spectral level repulsion, a hallmark of quantum chaos.