Recent developments in Quantum Spin Hall (QSH) effect have triggered much attention in inverted InAs/GaSb Quantum wells (QWs), which are the leading material in QSH systems. Inverted InAs/GaSb QWs are a type II heterostructure with the broken gap, where two dimensional (2D) electrons and holes are confined in spatially separated QWs. From the 1970s until now, the ground state of this structure has been discussed between two candidates: exciton insulator (BCS type exciton condensation) and hybridization gap. The QSH effect was theoretically proposed in the bulk hybridization gap. Although pioneer works about QSH effect have been performed, the conductive hybridization gap limits further exploration. For example, the existence of the QSH effect in this system is still not conclusive. In this thesis, through double-gate modulation, we investigated the whole phase in the inverted band of this structure. We observed two distinct quantum phases: time reversal symmetry (TRS) QSH insulator in the deeply inverted regime and exciton insulator in the shallowly inverted regime. In the deeply inverted regime, with the strain effect in InGaSb QW, we realized the insulating hybridization gap for the first time, which gave us the opportunity to observe TRS QSH effect in this system for the first time. With the largest bulk gap in known QSH systems, we observed the helical edges had the longest coherence length (nearly 13μm) and were more stable against temperature, compared with previous results, which paved the way to construct the room temperature topological circuit. In the shallowly inverted regime, the quantized plateau of QSH effect was observed in mesoscopic devices for the first time. Surprisingly, this helical edge mode was robust under the high magnetic field, demonstrating the first TRS broken QSH insulator. This novel quantum phase could not be understood in the single particle topological theory. Further studies showed that the bulk gap was dominated by exciton gap instead of hybridization gap. We performed the low temperature transport and Terahertz transmission measurement on the bulk exciton gap, and observed the solid evidence for the existence of BCS-like exciton condensation, which was under search for more than fifty years. Furthermore, we performed one dimensional Coulomb drag experiments in the topological circuit. We observed positive and negative drag results dependent on the temperature, indicating the charge symmetry and many-body correlation.