  Quantum droplets (QDs) are self-trapped modes stabilized by the
Lee-Huang-Yang correction to the mean-field Hamiltonian of binary atomic
Bose-Einstein condensates. The existence and stability of quiescent and
rotating dipole-shaped and vortex QDs with vorticity $S=1$ (DQDs and VQDs,
respectively) are numerically studied in the framework of the accordingly
modified two-component system. The rotating DQDs trapped in an annular
potential are built of two crescent-like components, stretching along the
azimuthal direction with the increase of the rotation frequency. Rotating
quadrupole QDs (QQDs) bifurcate from the VQDs with $S=2$. Above a certain
rotation frequency, they transform back into VQDs with a flat-top shape.
Rotating DQDs and QQDs are stable in a broad interval of values of the chemical
potential. The results provide the first example of stable modes which are
intermediate states between the rotating DQDs and QQDs on the one hand, and
VQDs on the other.
