  We present the model of a quantum dot (QD) consisting of a spherical
core-bulk heterostructure made of 3D topological insulator (TI) materials, such
as PbTe/Pb$_{0.31}$Sn$_{0.69}$Te, with bound massless and helical Weyl states
existing at the interface and being confined in all three dimensions. The
number of bound states can be controlled by tuning the size of the QD and the
magnitude of the core and bulk energy gaps, which determine the confining
potential. We demonstrate that such bound Weyl states can be realized for QD
sizes of few nanometers. We identify the spin locking and the Kramers pairs,
both hallmarks of 3D TIs. In contrast to topologically trivial semiconductor
QDs, the confined massless Weyl states in 3D TI QDs are localized at the
interface of the QD and exhibit a mirror symmetry in the energy spectrum. We
find strict optical selection rules satisfied by both interband and intraband
transitions that depend on the polarization of electron-hole pairs and
therefore give rise to the Faraday effect due to Pauli exclusion principle. We
show that the semi-classical Faraday effect can be used to read out spin
quantum memory. When a 3D TI QD is embedded inside a cavity, the single-photon
Faraday rotation provides the possibility to implement optically mediated
quantum teleportation and quantum information processing with 3D TI QDs, where
the qubit is defined by either an electron-hole pair, a single electron spin,
or a single hole spin in a 3D TI QD. Remarkably, the combination of inter- and
intraband transition gives rise to a large dipole moment of up to 450 Debye.
Therefore, the strong-coupling regime can be reached for a cavity quality
factor of $Q\approx10^{4}$ in the infrared wavelength regime of around
$10\:\mu$m.
