I'm interested in the derivation of the classical Hall effect coefficient, given in cgs by $$R_{H}=-\frac{1}{nec},$$ where $n$ is the electron number density, $-e<0$ is the electron charge,and $c$ is the usual, ubiquitous velocity in Physics, from the fact that QHE provides the quantum of electrical conductance $$g=\frac{2e^{2}}{h},$$ where $h$ is Planck's constant, and the 2 comes from spin degeneracy.
Is there a convenient way to go from the quantum to the classical case for this problem?
I think you cannot derive classical case from the quantum case. $g=\text{filling factor}\cdot\frac{2e^2}{h}$ occurs at very high magnetic fields where Landau levels start filling and current is carried only by edge states. In classical regime, magnetic fields are low and Landau levels haven't started filling and there is no edge states
If you consider a sample infinite in all directions (no edge states), you will get conventional quantum Hall effect (although it is not classical, since we deal with the Landau levels). This would however require some kind of Kubo-like calculation, so that the current is finite - this is not the case in QHE, where the resistance is finite due to the presence of the confining potential (note that IQHE is but conductance quantization in the quantum point contacts, formed by the sample edges and the magnetic field).
