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We present the electron tunneling transport and spectroscopic characters of a superconducting (SC) Josephson junction with a barrier of a single anisotropic Kitaev quantum spin liquid (QSL) layer. We find that the dynamical spin-correlation features are well reflected in the direct-current differential conductance dI^{c}/dV of single-particle tunneling, including the unique spin gap and dressed itinerant Majorana dispersive band, in addition to an energy shift 2Δ from the two-SC-lead gap. From the spectral characters, we identify different topological quantum phases of the anisotropic Kitaev QSL. We also present the zero-voltage Josephson current I^{s}, which displays residual features of the anisotropic Kitaev QSL. These results pave a new path to measurement of dynamical spinon or Majorana fermion spectroscopy of Kitaev and other spin-liquid materials.