Graphene-based allotropes such as carbon nanorings hold the promise of completely new nanodevice and metamaterials applications due to the effects of magnetic flux and curvature on quantum transport on a nanoscale toroidal surface and the coherence of resulting electromagnetic moments. Unique electronic and optical characteristics will emerge due to the compactification of the honeycomb lattice structure of a flat graphene sheet to a two-diemsional mamifold with toroidal geometry. Additional modular symmetries are predicted to significantly impact energy band structure and transport properties of physically distinct nanotori with different chiralities and dimensions and thus drastically reduce the number of spectrally distinct ring geometries. In addition to persistent current and Aharonov-Bohm effects under magnetic flux, new electromagnetic field distributions such as a new toroidal moment will be generated by the ring currents. In a metamaterial of a regular two- or three-dimensional lattice of these aligned nanoconstituents a significant enhancement of these quantum signatures may be expected coherence of the individual electromagnetic responses. In an effective model, the Hamiltonian for a single charge constrained to motion near a toroidal helix with loops of arbitrary eccentricity is developed and the resulting three-dimensional Schrödinger equation reduced to an effective one-dimensional formula inclusive of curvature effects in form of two resulting effective curvature potentials. The magnitude of the toroidal moment generated by the current depends strongly on the magnetic field component of the field normal to the toroidal plane. A strong dependence on coil eccentricity is also observed. In a theoretical sense, the curvature potential terms are necessary to preserve the hermiticity of the minimal prescription Hamiltonian. This effective model may also elucidate how a surface current may be driven by a properly polarized incoming electromagnetic wave front to generate a specific multipole response. Alternatively, electron transport on the carbon nanotorus is calculated in a tightbinding model for armchair and zigzag carbon nanotori between metallic leads using a recursive non-equilibrium Green's function method. Density-of-states, transmission function and source drain current are calculated for realistic system sizes of 10,000 carbon atoms and more. An object-oriented C++ code was developed using parallel sparse matrix software libraries such as PETSc (Portable, Extensible Toolkit for Scientific Computation) with additional MPI parallelism to evaluate the transport Green.s function at different energies. This fast and numerically precise tool on a multi-core architecture can incorporate additional effects such as electron-phonon coupling effects due to low-energy phonon modes, exciton transport, or electron-plasmon coupling terms in second- or third-nearest-neighbor type calculations.