This paper presents a theoretical prediction of a dissipationless quantum spin current at room temperature in hole-doped semiconductors such as Si, Ge, and GaAs. The spin current is induced by an electric field and is based on a generalization of the quantum Hall effect. The key idea is that the quantum Hall conductance is determined by the equilibrium response of all states below the Fermi level, and is independent of scattering rates. This leads to a non-dissipative transport of spin current, which can be described by a non-dissipative transport equation. The topological origin of the quantum Hall effect is revealed through the first Chern number of a U(1) gauge connection in momentum space. The predicted effect enables efficient spin injection without the need for metallic ferromagnets, and could lead to quantum spintronic devices with integrated information processing and storage units, operating with low power consumption and performing reversible quantum computation. The work is driven by the confluence of the important technological goals of quantum spintronics with the quest of generalizing the quantum Hall effect to higher dimensions. The quantum Hall effect is a manifestation of quantum mechanics observable at macroscopic scales. The Hall conductance in the quantum Hall effect is quantized and completely independent of any scattering rates in the system. The non-dissipative quantum Hall conductance is expressed in terms of equilibrium response of all states below the Fermi level. The topological origin of the quantum Hall effect is revealed through the fact that the Hall conductance can also be expressed as the first Chern number of a U(1) gauge connection defined in momentum space. Recently, the quantum Hall effect has been generalized to four spatial dimensions. In that case, an electric field induces an SU(2) spin current through the non-dissipative transport equation. The quantum Hall response in that system is physically realized through the spin-orbit coupling in a time-reversal symmetric system. At the boundary of this four-dimensional quantum liquid, when both the electric field and the spin current are restricted to the three-dimensional subspace, the dissipationless response is given by a specific equation. The paper also discusses the realization of this electric field-induced topological spin current in conventional hole-doped semiconductors. The effective Luttinger Hamiltonian for holes is given, and the eigenvalues are classified by the helicity. The paper then discusses the effect of a uniform electric field on the system, and the resulting non-trivial adiabatic gauge connection. The non-Abelian structure is only present in the light-hole band. The paper also discusses the correction due to the non-Abelian nature of the gauge connection of the light-hole band, and the resulting spin current. The spin current is found to be rotationally invariant and has a topological character. The spin conductivity is independent of the mean free path and relaxational rates, and all states below the Fermi energy contribute to theThis paper presents a theoretical prediction of a dissipationless quantum spin current at room temperature in hole-doped semiconductors such as Si, Ge, and GaAs. The spin current is induced by an electric field and is based on a generalization of the quantum Hall effect. The key idea is that the quantum Hall conductance is determined by the equilibrium response of all states below the Fermi level, and is independent of scattering rates. This leads to a non-dissipative transport of spin current, which can be described by a non-dissipative transport equation. The topological origin of the quantum Hall effect is revealed through the first Chern number of a U(1) gauge connection in momentum space. The predicted effect enables efficient spin injection without the need for metallic ferromagnets, and could lead to quantum spintronic devices with integrated information processing and storage units, operating with low power consumption and performing reversible quantum computation. The work is driven by the confluence of the important technological goals of quantum spintronics with the quest of generalizing the quantum Hall effect to higher dimensions. The quantum Hall effect is a manifestation of quantum mechanics observable at macroscopic scales. The Hall conductance in the quantum Hall effect is quantized and completely independent of any scattering rates in the system. The non-dissipative quantum Hall conductance is expressed in terms of equilibrium response of all states below the Fermi level. The topological origin of the quantum Hall effect is revealed through the fact that the Hall conductance can also be expressed as the first Chern number of a U(1) gauge connection defined in momentum space. Recently, the quantum Hall effect has been generalized to four spatial dimensions. In that case, an electric field induces an SU(2) spin current through the non-dissipative transport equation. The quantum Hall response in that system is physically realized through the spin-orbit coupling in a time-reversal symmetric system. At the boundary of this four-dimensional quantum liquid, when both the electric field and the spin current are restricted to the three-dimensional subspace, the dissipationless response is given by a specific equation. The paper also discusses the realization of this electric field-induced topological spin current in conventional hole-doped semiconductors. The effective Luttinger Hamiltonian for holes is given, and the eigenvalues are classified by the helicity. The paper then discusses the effect of a uniform electric field on the system, and the resulting non-trivial adiabatic gauge connection. The non-Abelian structure is only present in the light-hole band. The paper also discusses the correction due to the non-Abelian nature of the gauge connection of the light-hole band, and the resulting spin current. The spin current is found to be rotationally invariant and has a topological character. The spin conductivity is independent of the mean free path and relaxational rates, and all states below the Fermi energy contribute to the