  Inspired by the boolean discrepancy problem, we study the following
optimization problem which we term \textsc{Spherical Discrepancy}: given $m$
unit vectors $v_1, \dots, v_m$, find another unit vector $x$ that minimizes
$\max_i \langle x, v_i\rangle$. We show that \textsc{Spherical Discrepancy} is
APX-hard and develop a multiplicative weights-based algorithm that achieves
optimal worst-case error bounds up to lower order terms. We use our algorithm
to give the first non-trivial lower bounds for the problem of covering a
hypersphere by hyperspherical caps of uniform volume at least
$2^{-o(\sqrt{n})}$. We accomplish this by proving a related covering bound in
Gaussian space and showing that in this \textit{large cap regime} the bound
transfers to spherical space. Up to a log factor, our lower bounds match known
upper bounds in the large cap regime.
