Although absence of the local order parameters is a fundamental feature of the topological phases, the Berry connection that represents a quantum interference of many body states successfully characterizes the bulk of the topological phases. The Chern number for the bulk of the quantum Hall states is a typical example. Also, with boundaries, appearance of local modes as the edge states characterizes the phase as the bulk-edge correspondence. As for the short range entangled state, the Berry phase that is quantized due to symmetry is used as a “quantum” local order parameter of the bulk. Z_2 Berry phase for the Haldane phase of the spin 1 quantum spin chain is a typical example. Here again, with boundaries, low energy local modes as the edge states appear associated with the nontrivial Berry phases as the bulk-edge correspondence. Focusing on the systems with interaction, we demonstrate the use of the symmetry protected Berry phases and the bulk-edge correspondence for various examples such as the generic valence bond solid (VBS) states and the corner states of the higher order topological phases.