  I perform a complete classification of 2d, quasi-1d and 1d topological
superconductors which originate from the suitable combination of inhomogeneous
Rashba spin-orbit coupling, magnetism and superconductivity. My analysis
reveals alternative types of topological superconducting platforms for which
Majorana fermions are accessible. Specifically, I observe that for quasi-1d
systems with Rashba spin-orbit coupling and time-reversal violating
superconductivity, as for instance due to a finite Josephson current flow,
Majorana fermions can emerge even in the absence of magnetism. Furthermore, for
the classification I also consider situations where additional "hidden"
symmetries emerge, with a significant impact on the topological properties of
the system. The latter, generally originate from a combination of space group
and complex conjugation operations that separately do not leave the Hamiltonian
invariant. Finally, I suggest alternative directions in topological quantum
computing for systems with additional unitary symmetries.
