  We present results for the $x$ dependence of the unpolarized, helicity, and
transversity isovector quark distributions in the proton using lattice QCD,
employing the method of quasi-distributions proposed by Ji in 2013. Compared to
a previous calculation by us, the errors are reduced by a factor of about 2.5.
Moreover, we present our first results for the polarized sector of the proton,
which indicate an asymmetry in the proton sea in favor of the $u$ antiquarks
for the case of helicity distributions, and an asymmetry in favor of the $d$
antiquarks for the case of transversity distributions.
