  We prove endpoint-type sparse bounds for Walsh-Fourier Marcinkiewicz
multipliers and Littlewood-Paley square functions. These results are motivated
by conjectures of Lerner in the Fourier setting. As a corollary, we obtain
novel quantitative weighted norm inequalities for these operators. Among these,
we establish the sharp growth rate of the $L^p$ weighted operator norm in terms
of the $A_p$ characteristic in the full range $1<p<\infty$ for
Walsh-Littlewood-Paley square functions, and a restricted range for
Marcinkiewicz multipliers. Zygmund's $L{(\log L)^{{\frac12}}}$ inequality is
the core of our lacunary multi-frequency projection proof. We use the Walsh
setting to avoid extra complications in the arguments.
