  We study the Vapnik-Chervonenkis (VC) dimension of the set of quadratic
residues (i.e. squares) in finite fields, $\mathbb F_q$, when considered as a
subset of the additive group. We conjecture that as $q \to \infty$, the squares
have the maximum possible VC-dimension, viz. $(1+o(1))\log_2 q$. We prove,
using the Weil bound for multiplicative character sums, that the VC-dimension
is $\geq (\frac{1}{2} + o(1))\log_2 q$. We also provide numerical evidence for
our conjectures. The results generalize to multiplicative subgroups $\Gamma
\subseteq \mathbb F_q^\times$ of bounded index.
