We analyze the dynamics of quantum supercooled liquids in terms of tagged particle dynamics. Unlike the classical case, uncertainty in the position of a particle in quantum liquid leads to qualitative changes. We demonstrate these effects in the dynamics of the first two moments of displacements, namely, the mean-squared displacement, 〈Δr^{2}(t)〉, and 〈Δr^{4}(t)〉. Results are presented for a hard-sphere liquid using mode-coupling theory formulation and simulation on a binary Lennard-Jones liquid. As the quantumness (controlled by the de Broglie thermal wavelength) is increased, a nonzero value of the moments at zero time leads to significant deviations from the classical behavior in the initial dynamics. Initial displacement shows ballistic behavior 〈Δr^{2}(t)〉∼t^{2}, but, as a result of large uncertainty in the position, the dynamical effects become weaker with increasing quantumness over this timescale.