(I don't know if there are standard names for this, so) Let's say that a Nondeterministic Finite Automaton (NFA) is $n$-expansive if it has $n$ states and any Deterministic Finite Automaton (DFA) recognizing the same language has at least $2^{n/2}$ states.
Let $E_n$ be the set of all $n$-expansive NFAs.
Is there a characterization to $E_n$?
Given any NFA $A$, is there a way to test if $A \in E_n$ in polynomial time in $n$ (therefore, without constructing the minimal DFA of $A$)?
The lower bound $2^{n/2}$ is arbitrary. If you know a characterization for other similar exponential bounds (e.g. $2^{n-1}$ or $2^{n/10}$), it will be very helpful.