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We investigate the physical 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 reconstruct the Dirac-Bergmann analysis and 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, we perform the analysis of coincident $f(Q)$-gravity. We find that 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 physical degrees of freedom of the theory and there are no longer any remaining gauge degrees of freedom.