  In a recent work, O'Donnell, Servedio and Tan (STOC 2019) gave explicit
pseudorandom generators (PRGs) for arbitrary $m$-facet polytopes in $n$
variables with seed length poly-logarithmic in $m,n$, concluding a sequence of
works in the last decade, that was started by Diakonikolas, Gopalan, Jaiswal,
Servedio, Viola (SICOMP 2010) and Meka, Zuckerman (SICOMP 2013) for fooling
linear and polynomial threshold functions, respectively. In this work, we
consider a natural extension of PRGs for intersections of positive
spectrahedrons. A positive spectrahedron is a Boolean function
$f(x)=[x_1A^1+\cdots +x_nA^n \preceq B]$ where the $A^i$s are $k\times k$
positive semidefinite matrices. We construct explicit PRGs that $\delta$-fool
"regular" width-$M$ positive spectrahedrons (i.e., when none of the $A^i$s are
dominant) over the Boolean space with seed length $\textsf{poly}(\log k,\log n,
M, 1/\delta)$.
  Our main technical contributions are the following: We first prove an
invariance principle for positive spectrahedrons via the well-known Lindeberg
method. As far as we are aware such a generalization of the Lindeberg method
was unknown. Second, we prove various geometric properties of positive
spectrahedrons such as their noise sensitivity, Gaussian surface area and a
Littlewood-Offord theorem for positive spectrahedrons. Using these results, we
give applications for constructing PRGs for positive spectrahedrons, learning
theory, discrepancy sets for positive spectrahedrons (over the Boolean cube)
and PRGs for intersections of structured polynomial threshold functions.
