  Restarting a deterministic process always impedes its completion. However, it
is known that restarting a random process can also lead to an opposite outcome
-- expediting completion. Hence, the effect of restart is contingent on the
underlying statistical heterogeneity of the process' completion times. To
quantify this heterogeneity we bring a novel approach to restart: the
methodology of inequality indices, which is widely applied in economics and in
the social sciences to measure income and wealth disparity. Using this approach
we establish an `inequality roadmap' for the mean-performance of sharp restart:
a whole new set of universal inequality criteria that determine when restart
with sharp timers (i.e. with fixed deterministic timers) decreases/increases
mean completion. The criteria are based on a host of inequality indices
including Bonferroni, Gini, Pietra, and other Lorenz-curve indices; each index
captures a different angle of the restart-inequality interplay. Utilizing the
fact that sharp restart can match the mean-performance of any general restart
protocol, we prove -- with unprecedented precision and resolution -- the
validity of the following statement: restart impedes/expedites mean completion
when the underlying statistical heterogeneity is low/high.
