  We construct small-$x$ evolution equations which can be used to calculate
quark and anti-quark helicity TMDs and PDFs, along with the $g_1$ structure
function. These evolution equations resum powers of $\alpha_s \, \ln^2 (1/x)$
in the polarization-dependent evolution along with the powers of $\alpha_s \,
\ln (1/x)$ in the unpolarized evolution which includes saturation effects. The
equations are written in an operator form in terms of polarization-dependent
Wilson line-like operators. While the equations do not close in general, they
become closed and self-contained systems of non-linear equations in the
large-$N_c$ and large-$N_c \, \& \, N_f$ limits. We construct a numerical
solution of the helicity evolution equations in the large-$N_c$ limit.
Employing the extracted intercept, we are able to predict directly from theory
the behavior of the quark helicity PDFs at small $x$, which should have
important phenomenological consequences. We also give an estimate of how much
of the proton's spin may be at small $x$ and what impact this has on the
so-called "spin crisis."
  Based on JHEP 1601 (2016) 072 (arXiv:1511.06737), arXiv:1610.06197 and
arXiv:1610.06188.
