We experimentally demonstrate a dynamical classification approach for investigation of topological quantum phases using a solid-state spin system through nitrogen-vacancy (NV) center in diamond. Similar to the bulkboundary correspondence in real space at equilibrium, we observe a dynamical bulk-surface correspondence in the momentum space from a dynamical quench process. An emergent dynamical topological invariant is precisely measured in experiment by imaging the dynamical spin-textures on the recently defined band-inversion surfaces, with high topological numbers being implemented. Importantly, the dynamical classification approach is shown to be independent of quench ways and robust to the decoherence effects, offering a novel and practical strategy for dynamical topology characterization, especially for high dimensional gapped topological phases.The topology of quantum systems has been developed into a major focus of research in physics since the discovery of quantum Hall effect [1,2]. Beyond the states of quantum matter characterized by Landau symmetry-breaking theory, the topological quantum phases bear a myriad of properties depending only on the topology [3-5], with the most celebrated paradigms discovered recently including the topological insulators [6][7][8][9][10][11] and semimetals [12,13]. Among the many exotic features emerging in a topological matter, the bulk-boundary correspondence is the most fundamental phenomenon showing that on the boundary gapless states can be obtained corresponding to and protected by the nontrivial topology in the bulk [14,15].At equilibrium, the number of topologically protected surface (edge) states is uniquely related to the bulk topological invariants [14,15]. This allows to detect topological insulators [8][9][10][11] and semimetals [12,13] by resolving the surface states from transport measurement or angle resolved photoemission spectroscopy. Apart from the direct measurement in condensed matter physics, quantum simulation may provide new strategies for the characterization [16][17][18][19][20][21][22][23][24][25][26][27][28][29]. For example, the band topology of 1D Su-Schrieffer-Heeger (SSH) chain can be determined by measuring the Zak phase [19,20], the bulk topology of a 2D Chern insulator can be observed by Hall transport studies [21,22], by Berry curvature mapping [27], or by a minimal measurement strategy [28] of imaging the spin texture at symmetric Bloch space [29]. In addition, these systems naturally offer powerful probes and controllability in comparison with their condensed matter counterpart, which enables the study of non-equilibrium physics across topological phase transitions. A generic protocol in state-of-the-art experiments is to prepare a topologically trivial initial state and then observe the non-equilibrium dynamics after the Hamiltonian has been quenched into a topological regime. For such non-equilibrium process, the bulk-boundary correspondence, however, is no-longer valid [30][31][32][33]. A natural challenge is to find a generic method ...