Heterointerfaces with symmetry breaking and strong interfacial coupling could give rise to the enormous exotic quantum phenomena. Here, we report on the experimental observation of intriguing two-dimensional superconductivity with superconducting transition temperature ($T_c$) of 3.8 K at heterostructure of Mott insulator Ti$_2$O$_3$ and polar semiconductor GaN revealed by the electrical transport and magnetization measurements. Furthermore, at the verge of superconductivity we find a wide range of temperature independent resistance associated with vanishing Hall resistance, demonstrating the emergence of quantum metallic-like state with the Bose-metal scenario of the metallic phase. By tuning the thickness of Ti$_2$O$_3$ films, the emergence of quantum metallic-like state accompanies with the appearance of superconductivity as decreasing in temperature, implying that the two-dimensional superconductivity is evolved from the quantum metallic-like state driven by the cooperative effects of the electron correlation and the interfacial coupling between Ti$_2$O$_3$ and polar GaN. These findings provide a new platform for the study of intriguing two-dimensional superconductivity with a delicate interplay of the electron correlation and the interfacial coupling at the heterostructures, and unveil the clues of the mechanism of unconventional superconductivity.
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