We study supercooled dynamics in a quantum hard-sphere liquid using quantum mode-coupling formulation. In the moderate quantum regime, classical cage effects lead to slower dynamics compared to the strongly quantum regime, where tunneling overcomes classical caging, leading to faster relaxation. As a result, the glass transition critical density can become significantly higher than for the classical liquids. A perturbative approach is used to solve time dependent quantum mode-coupling equations to study in detail the dynamics of the supercooled liquid in the moderate quantum regime. Similar to the classical case, the relaxation time shows the power-law increase with the increase in the density in the supercooled regime. However, the power-law exponent is found to be dependent on the quantumness; it increases linearly as the quantumness is increased in the moderate quantum regime.