  We present the electron tunneling transport and spectroscopic characters of a
superconducting {\it Josephson} junction with a barrier of 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 the single-particle tunneling, including the unique
spin gap and dressed itinerant Majorana dispersive band, in addition to an
energy shift $2\Delta$ of two-lead superconducting gaps. From the spectral
characters, we identify different topological quantum phases of the anisotropic
Kitaev QSL. We also present the zero-voltage {\it Josephson} current $I^{s}$
which displays residual features of the anisotropic Kitaev QSL. These results
pave a new way to measure the dynamical spinon or Majorana fermion spectroscopy
of the Kitaev and other spin liquid materials.
