We have pointed out the possibility of quantum Hall effect or quantum patterns of transportation in a degenerate strongly magnetized quark matter, which might be expected inside a highly dense compact star. An anisotropic pattern of electrical conductivity and resistivity tensor in classical and quantum cases is explored by considering cyclotron motion and Landau quantization respectively. With increasing magnetic field, classical to quantum transitions are realized through enhanced/reduced resistivity/conductivity with jumping pattern. Considering QCD relaxation time scale of 10 fm, $eBapprox (1-4) m_pi^2$ might be considered as strong magnetic field for massless and degenerate quark matter with quark chemical potential $muapprox 0.2-0.4$ GeV. Beyond these threshold ranges of magnetic field, perpendicular motion of quarks might be stopped and 3 $rightarrow$ 1 dimensionally reduced conduction picture might be established.
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