  Dynamical measurement schemes are an important tool for the investigation of
quantum many-body systems, especially in the age of quantum simulation. Here,
we address the question whether generic measurements can be implemented
efficiently if we have access to a certain set of experimentally realizable
measurements and can extend it through time evolution. For the latter, two
scenarios are considered (a) evolution according to unitary circuits and (b)
evolution due to Hamiltonians that we can control in a time-dependent fashion.
We find that the time needed to realize a certain measurement to a predefined
accuracy scales in general exponentially with the system size -- posing a
fundamental limitation. The argument is based, on the construction of
$\varepsilon$-packings for manifolds of observables with identical spectra and
a comparison of their cardinalities to those of $\varepsilon$-coverings for
quantum circuits and unitary time-evolution operators. The former is related to
the study of Grassmann manifolds.
