  Searches for gravitational wave echoes in the aftermath of mergers and/or
formation of astrophysical black holes have recently opened a novel and
surprising window into the quantum nature of their horizons. Similar to astro-
and helioseismology, study of the spectrum of quantum black holes provides a
promising method to understand their inner structure, what we call
$\textit{quantum black hole seismology}$. We provide a detailed numerical and
analytic description of this spectrum in terms of the properties of the Kerr
spacetime and quantum black hole horizons, showing that it drastically differs
from their classical counterparts. Our most significant findings are the
following: (1) If the temperature of quantum black hole is $\lesssim 2 \times$
Hawking temperature, then it will not suffer from ergoregion instability
(although the bound is looser at smaller spins). (2) We find how quantum black
hole spectra pinpoint the microscopic properties of quantum structure. For
example, the detailed spacing of spectral lines can distinguish whether quantum
effects appear through compactness (i.e., exotic compact objects) or frequency
(i.e., modified dispersion relation). (3) We find out that the overtone
quasinormal modes may strongly enhance the amplitude of echo in the
low-frequency region. (4) We show the invariance of the spectrum under the
generalized Darboux transformation of linear perturbations, showing that it is
a genuine covariant observable.
