Since the notion of topological insulator (TI) was envisioned in late 2000s, topology has become a new paradigm in condensed matter physics. Realization of topology as a generic property of materials has led to numerous predictions of topological effects. Although most of the classical topological effects, directly resulting from the presence of the spin-momentum-locked topological surface states (TSS), were experimentally confirmed soon after the theoretical prediction of TIs, many topological quantum effects remained elusive for a long while. It turns out that native defects, particularly interfacial defects, have been the main culprit behind this impasse. Even after quantum regime is achieved for the bulk states, TSS still tends to remain in the classical regime due to high density of interfacial defects, which frequently donate mobile carriers due to the very nature of the topologically-protected surface states. However, with several defect engineering schemes that suppress these effects, a series of topological quantum effects have emerged including quantum anomalous Hall effect, quantum Hall effect, quantized Faraday/Kerr rotations, topological quantum phase transitions, axion insulating state, zeroth-Landau level state, etc. Here, we review how these defect engineering schemes have allowed topological surface states to pull out of the murky classical regime and reveal their elusive quantum signatures, over the past decade.