  The appearance of negative terms in quasiprobability representations of
quantum theory is known to be inevitable, and, due to its equivalence with the
onset of contextuality, of central interest in quantum computation and
information. Until recently, however, nothing has been known about how much
negativity is necessary in a quasiprobability representation. Zhu proved that
the upper and lower bounds with respect to one type of negativity measure are
saturated by quasiprobability representations which are in one-to-one
correspondence with the elusive symmetric informationally complete quantum
measurements (SICs). We define a family of negativity measures which includes
Zhu's as a special case and consider another member of the family which we call
"sum negativity." We prove a sufficient condition for local maxima in sum
negativity and find exact global maxima in dimensions $3$ and $4$. Notably, we
find that Zhu's result on the SICs does not generally extend to sum negativity,
although the analogous result does hold in dimension $4$. Finally, the Hoggar
lines in dimension $8$ make an appearance in a conjecture on sum negativity.
