The time coordinates $t_{\mathrm{L}}$ and $t_{\mathrm{R}}$ on the left and right boundaries both increase upward (or downward).

The characteristic timescale marking the onset of the linear ramp is the Thouless time $T_{\text{Th}}$ [4, 5, 6, 7, 8]. This timescale is observable-dependent (nonuniversal), and therefore the precise onset of the ramp in the generating functions of complexity studied here generally differs from that of the spectral form factor.

We refer to the measures of complexity for boundary field theory as quantum complexity.

In contrast to the spectral form factor [50], the generating functions of complexity $⟨e^{-\alpha \mathcal{C}}⟩$ defined in Eq. (9) are expected to be self-averaging in the large-entropy limit.

We adopt the convention for energy eigenstates as $⟨E_i|E_j⟩\equiv [\delta (E_{ij})/e^{S_0}\overline{D}(E_i)]$.

From the viewpoint of the dual boundary theory, the energy eigenstates $|E⟩$ can be expressed as products of left- and right-sided states, i.e., $|E⟩=|\mathbf{E}{⟩}_{\mathrm{L}}\otimes |\mathbf{E}{⟩}_{\mathrm{R}}$. In this Letter, however, we restrict ourselves to the diagonal subspace spanned by $|E⟩$ and do not explicitly utilize the product structure.

These generalized volume functionals naturally appear in the context of holographic complexity in higher-curvature theories of gravity; see, e.g., [54, 55].

The timescale inversely related to the Thouless energy $E_{\text{Th}}$ is the Thouless time $T_{\text{Th}}\sim E_{\text{Th}}^{-1}$.

For clarity, we begin with explicit results for the Gaussian unitary ensemble and later extend the analysis to more general conclusions without assuming a specific universality class.

The canonical version is given by adding a thermal factor $e^{-[\beta /2](E_1+E_2)}$.