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Quantum simulation, as a state-of-art technique, provides the powerful way to explore topological quantum phases beyond natural limits. Nevertheless, a previously-not-realized three-dimensional (3D) chiral topological insulator, and demonstrate by quantum quenches a complete study of both the bulk and surface topological physics. First, a dynamical bulk-surface correspondence in momentum space is observed, showing that the bulk topology of the 3D phase uniquely corresponds to the nontrivial quench dynamics emerging on 2D momentum hypersurfaces called band inversion surfaces (BISs), equivalent to the bulk-boundary correspondence in real space. Further, the symmetry protection of the 3D chiral phase is uncovered by measuring dynamical spin textures on BISs, which exhibit perfect (broken) topology when the chiral symmetry is preserved (broken). Finally we measure the topological charges to characterize directly the bulk topology, and identify an emergent dynamical topological transition when varying the quenches from deep to shallow regimes. This work opens a new avenue of quantum simulation towards for the complete study of topological quantum phases.