This manuscript describes my research on quantum spintronics (ballistic spin transport) based on 3D topological insulator nanowires. It served as the basis for the oral examination to get an accreditation to supervise research and PhD students from the Université Grenoble Alpes (UGA). My experimental research activities focus on spin-related quantum phenomena in magnets and low-dimensional conductors, mostly studied by electrical transport measurements at very low temperatures and in 3D vector magnetic fields. In particular, novel quantum electronic phases are investigated in nanostructures of topological materials, such as strong 3D topological insulators (3D TIs) induced by spin-orbit coupling, for which the influence of the geometric Berry phase can lead to spin-textured gapless quasi-particles. In this dissertation, a first part gives an exhaustive review of the results obtained on nanostructures of disordered 3D topological insulators, from nanoflakes to nanowires. Quantum transport measurements are used to evidence some unique properties of spin-helical 2D Dirac fermions, due to their anisotropic scattering by disorder. In quasi-1D quantum wires, quantum confinement in reduced dimension leads to quasi-ballistic transport, with a hollow-conductor geometry as for carbon nanotubes and semiconducting core-shell nanowires, but with strong disorder. The second part describes my on-going research project, with the aim to investigate 1D dissipationless spin-polarized chiral edge states. Two directions are followed : i) the search for the perfectly-transmitted mode in (Bi,Sb)Te nanowires or in core-shell 3D TI heterostructures ; ii) the search for the quantum anomalous Hall and axion insulator regimes (among other topological phases) in novel magnetic topological insulators of the MnBi2Te4 family, and related materials.