  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 spectrahedra via the well-known Lindeberg
method. As far as we are aware such a generalization of the Lindeberg method
was unknown. Second, we prove an upper bound on noise sensitivity and a
Littlewood-Offord theorem for positive spectrahedra. Using these results, we
give applications for constructing PRGs for positive spectrahedra, learning
theory, discrepancy sets for positive spectrahedra (over the Boolean cube) and
PRGs for intersections of structured polynomial threshold functions.
