  We study the convex hull of a set $S\subset \mathbb{R}^n$ defined by three
quadratic inequalities. A simple way of generating inequalities valid on $S$ is
to take nonnegative linear combinations of the defining inequalities of $S$. We
call such inequalities aggregations. We introduce a new technique relating
aggregations to properties of the spectral curve, i.e. the curve defined by the
vanishing of the determinant polynomial, and utilizing known spectral sequences
(Agrachev and Lerario, 2012). We find new families beyond those identified in
(Dey, Mu\~noz, and Serrano, 2022; Blekherman, Dey, and Sun, 2024), where the
convex hull is defined by aggregations. We also prove a characterization of the
emptiness of the projective variety defined by $3$ homogeneous quadratics in
terms of the spectral curve generalizing results of (Agrachev, 1988).
