We consider the existence of pairs of probability ensembles which
may be efficiently distinguished given $k$ samples
but cannot be efficiently distinguished given $k'<k$ samples.
It is well known that in any such pair of ensembles it cannot be that
both are efficiently computable
(and that such phenomena ... more >>>

We consider random walks on "balanced multislices"
of any "grid" that respects the "symmetries" of the grid, and show that a broad class of such walks are good spectral expanders. (A grid is a set of points of the form $\mathcal{S}^n$ for finite $\mathcal{S}$, and a balanced multi-slice is the ... more >>>