  We investigate the propagating degrees of freedom of $f(Q)$-gravity in a
$4$-dimensional space-time under the imposition of the coincident gauge by
performing the Dirac-Bergmann analysis. In this work, we start with a top-down
reconstruction of the metric-affine gauge theory of gravity based only on the
concept of a vector bundle. Then, the so-called geometrical trinity of gravity
is introduced and the role of the coincident GR is clarified. After that, we
reveal relationships between the boundary terms in the variational principle
and the symplectic structure of the theory in order to confirm the validity of
the analysis for our studied theories. Then, as examples, we revisit the
analysis of GR and its $f(\lc{R})$-extensions. Finally, after reviewing the
Dirac-Bergmann analysis of the coincident GR and that of $f(T)$-gravity, we
perform the analysis of coincident $f(Q)$-gravity. Under the imposition of
appropriate spatial boundary conditions, we find that, as a generic case, the
theory has five primary, three secondary, and two tertiary constraint densities
and all these constraint densities are classified into second-class constraint
density; the number six is the propagating degrees of freedom of the theory and
there are no longer any remaining gauge degrees of freedom. We also discuss the
condition of providing seven pDoF as a generic case. The violation of
diffeomorphism invariance of coincident $f(Q)$-gravity make it possible to
emerge such several sectors.
