  We show how to distinguish circuits with $\log k$ negations (a.k.a
$k$-monotone functions) from uniformly random functions in
$\exp\left(\tilde{O}\left(n^{1/3}k^{2/3}\right)\right)$ time using random
samples. The previous best distinguisher, due to the learning algorithm by
Blais, Cannone, Oliveira, Servedio, and Tan (RANDOM'15), requires
$\exp\big(\tilde{O}(n^{1/2} k)\big)$ time.
  Our distinguishers are based on Fourier analysis on \emph{slices of the
Boolean cube}. We show that some "middle" slices of negation-limited circuits
have strong low-degree Fourier concentration and then we apply a variation of
the classic Linial, Mansour, and Nisan "Low-Degree algorithm" (JACM'93) on
slices. Our techniques also lead to a slightly improved weak learner for
negation limited circuits under the uniform distribution.
