  We present the electron tunneling transport and spectroscopic characters of a
superconducting Josephson junction with a barrier of single 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 with an energy shift of superconducting gap sum
2{\Delta}, including the unique spin gap and dressed itinerant Majorana
dispersive band, which can be regarded as evidence of the Kitaev QSL. However,
the zero-voltage Josephson current I_s only displays some residual features of
dynamical spin susceptibility in the Kitaev QSL due to the spin singlet of
Cooper pairs. These results pave a new way to measure the dynamical spinon or
Majorana fermion spectroscopy of the Kitaev and other spin liquid materials.
