  We present the evolution of the mass-metallicity (MZ) relations at z=4-10
derived with 135 galaxies identified in the JWST/NIRSpec data taken from the
three major public spectroscopy programs of ERO, GLASS, and CEERS. Because
there are many discrepancies between flux measurements reported by early ERO
studies, we first establish our NIRSpec data reduction procedure for reliable
emission-line flux measurements and errors successfully explaining Balmer
decrements with no statistical tensions via thorough comparisons of the early
ERO studies. Applying the reduction procedure to the 135 galaxies, we obtain
emission-line fluxes for physical property measurements. We confirm that 10 out
of the 135 galaxies with [OIII]4363-lines have electron temperatures of
(1.1-2.3)*10^4K, similar to lower-z star-forming galaxies, that can be
explained by heating of young massive stars. We derive metallicities of the 10
galaxies by the direct method and the rest of the galaxies with strong lines by
the metallicity calibrations of Nakajima et al. (2022) applicable for these
low-mass metal-poor galaxies, anchoring the metallicities with the
direct-method measurements. We thus obtain MZ relations and star-formation rate
(SFR)-MZ relations over z=4-10. We find that there is a small evolution of the
MZ relation from z~2-3 to z=4-10, while interestingly that the SFR-MZ relation
shows no evolution up to z~8 but a significant decrease at z>8 beyond the
error. This SFR-MZ relation decrease at z>8 may suggest a break of the
metallicity equilibrium state via star-formation, inflow, and outflow, while
further statistical and local-baseline studies are needed for a conclusion.
