  Higher-order topological insulators (HOTIs) are systems with topologically
protected in-gap boundary states localized at their $(d-n)$-dimensional
boundaries, with $d$ the system dimension and $n$ the order of the topology.
This work proposes a rather universal dynamics-based characterization of one
large class of $Z$-type HOTIs without specifically relying on any symmetry
considerations. The key element of our innovative approach is to connect
quantum quench dynamics with nested configurations of the so-called band
inversion surfaces (BISs) of momentum-space Hamiltonians as a sum of operators
from the Clifford algebra (a condition that can be relaxed), thereby making it
possible to dynamically detect each and every order of topology on an equal
footing. Given that experiments on synthetic topological matter can directly
measure the winding of certain pseudospin texture to determine topological
features of BISs, the topological invariants defined through nested BISs are
all within reach of ongoing experiments. Further, the necessity of having
nested BISs in defining higher-order topology offers a unique perspective to
investigate and engineer higher-order topological phase transitions.
