  We show that for every fixed $k\geq 3$, the problem whether the
termination/counter complexity of a given demonic VASS is $\mathcal{O}(n^k)$,
$\Omega(n^{k})$, and $\Theta(n^{k})$ is coNP-complete, NP-complete, and
DP-complete, respectively. We also classify the complexity of these problems
for $k\leq 2$. This shows that the polynomial-time algorithm designed for
strongly connected demonic VASS in previous works cannot be extended to the
general case. Then, we prove that the same problems for VASS games are
PSPACE-complete. Again, we classify the complexity also for $k\leq 2$.
Interestingly, tractable subclasses of demonic VASS and VASS games are obtained
by bounding certain structural parameters, which opens the way to applications
in program analysis despite the presented lower complexity bounds.
