  Topological phase of matter is now a mainstream of research in condensed
matter physics, of which the classification, synthesis, and detection of
topological states have brought excitements over the recent decade while remain
incomplete with ongoing challenges in both theory and experiment. Here we
propose to establish a universal dynamical characterization of the topological
quantum phases classified by integers, and further propose the high-precision
dynamical schemes to detect such states. The framework of {the dynamical
classification theory} consists of basic theorems. First, we uncover that
classifying a $d$-dimensional ($d$D) gapped topological phase {of generic
multibands} can reduce to a ($d-1$)D invariant defined on so-called band
inversion surfaces (BISs), rendering a {\it bulk-surface duality} which
simplifies the topological characterization. Further, we show in quenching
across phase boundary the (pseudo)spin dynamics to exhibit unique topological
patterns on BISs, which are attributed to the post-quench bulk topology and
manifest a {\it dynamical bulk-surface correspondence}. For this the
topological phase is classified by a dynamical topological invariant measured
from dynamical spin-texture field on the BISs. Applications to quenching
experiments on feasible models are proposed and studied, demonstrating the new
experimental strategies to detect topological phases with high feasibility.
This work opens a broad new direction to classify and detect topological phases
by quantum dynamics.
