Pole skipping of bulk higher-spin fields in AdS black holes has been previously studied in [18]. Owing to the highly involved nature of the bulk equations, only partial results were obtained. We will compare our findings with theirs below.

We relegate the details to Sec. S-I of Supplemental Material [34] and present only the final results here.

Although there is no translation symmetry along the radial direction in hyperbolic space, it is nevertheless possible to define $k$; see Sec. S-I of Supplemental Material [34].

There are also a few pole skipping points that location depend on the specific operator choice in a nonuniversal way. We will neglect them. See Sec. S-I of Supplemental Material [34].

Note that [18] does not obtain all the pole skipping points for each $n$.

More generally, one can define the velocity-dependent Lyapunov exponent $\lambda (v)$ [52] for OTOC with large spacetime separation with $v=|\rho /T|$. It is given by the saddle-pole transition on the leading Regge trajectory [29]. In the pole-dominant regime, the velocity-dependent Lyapunov exponent is ballistic $\lambda (v)=2\pi /\beta (1-v/v_B)$, where $v_B$ is the butterfly velocity. The pole-dominant regime can be determined by the pole skipping of stress tensor at (4) with $J=2,\mathrm{\Delta }=d$. Therefore, all quantum chaotic data of OTOC can be extracted from pole skipping.

We mention by passing that it appears natural to conjecture that the first line of (6) corresponds to “descendants” of the highest pole skipping points, arising from the same underlying horizon symmetries, while the second line originates from a different physical mechanism.

For the highest pole skipping points, an explanation for this universality is their origin from horizon symmetries.

In Sec. S-II of Supplemental Material [34], we give a precise definition of composite operators in a generic theory and study their pole skipping behavior in the Rindler CFT, where additional pole skipping points are observed, potentially attributable to nonlocal intermediate operators. We leave elucidation of their origin to future work.

We should choose the solution of $h\in (1,2]$ for $p\in [-\pi ,\pi ]$.

$j_{\text{leading}}(p)-1=[h(p)-1]{\lambda }_0$, which is also precisely the momentum-dependent Lyapunov exponent discussed by [26].