Given two *n*-element
structures, *A* and *B*, which can be distinguished by a
sentence of *k*-variable
first-order logic, what is the minimum *f*(*n*,*k*) such that
there is guaranteed to be a sentence *ϕ* ∈ ℒ_{k} with at
most *f*(*n*,*k*)
quantifiers, such that *A* ⊨ *ϕ* but *B* ⊭ *ϕ*? We will present
various results related to this question obtained by using the recently
introduced QVT games. Through the lens of this question, we will
highlight some difficulties that arise in analysing the QVT game and
share some insights which can help to overcome them. A core theme is
that of lifting quantifier depth lower bounds to quantifier number lower
bounds.