  This work theoretically investigates \textcolor{black}{the stationary
properties} and the dynamics of the rotating quantum liquid droplets confined
in a two-dimensional symmetric anharmonic trap. Mimicking the quantum Hall
systems, the modified Gross-Pitaevskii equation with the inclusion of the
Lee-Huang-Yang nonlinear interaction is analytically solved, and the role of
the Landau-level mixing effect is addressed. \textcolor{black}{Via controlling
the nonlinear interaction and the rotation speed, the rotating quantum droplet
with multiply quantized vortex can be created, and the preference of the
energetically favored quantum states can be distinguished in the phase diagram.
To better interpret the underlying physics of the phase singularities, a brief
comparison of the rotating quantum droplet and the optical vortex is made. The
investigation of the long-term evolution of the rotating quantum droplets
confirms the stability of the quantum states. At certain rotation speeds, the
multi-periodic trajectories and breathings provide evidence of the emergence of
the collective excitation of the surface mode in the vortex state. For quantum
droplets carrying multiply quantized vortex, the microscopic snapshots of the
rotation field adjusted current density distribution show that the combined
nonlinear interaction and the anharmonic trapping potential can provide the
restoring force to lead the quantum droplet to a regular and stable revolution
and reach the dynamic equilibrium, revealing the signature of the generation of
superfluids in the new kind of low-dimensional quantum liquids.
