  In 1981, Neil Immerman described a two-player game, which he called the
"separability game" \cite{Immerman81}, that captures the number of quantifiers
needed to describe a property in first-order logic. Immerman's paper laid the
groundwork for studying the number of quantifiers needed to express properties
in first-order logic, but the game seemed to be too complicated to study, and
the arguments of the paper almost exclusively used quantifier rank as a lower
bound on the total number of quantifiers. However, last year Fagin, Lenchner,
Regan and Vyas rediscovered the games, provided some tools for analyzing them,
and showed how to utilize them to characterize the number of quantifiers needed
to express linear orders of different sizes. In this paper, we push forward in
the study of number of quantifiers as a bona fide complexity measure by
establishing several new results. First we carefully distinguish minimum number
of quantifiers from the more usual descriptive complexity measures, minimum
quantifier rank and minimum number of variables. Then, for each positive
integer $k$, we give an explicit example of a property of finite structures (in
particular, of finite graphs) that can be expressed with a sentence of
quantifier rank $k$, but where the same property needs $2^{\Omega (k^2)}$
quantifiers to be expressed.
