  We perform the ADM decomposition of a five-parameter family of quadratic
non-metricity theories and study their conjugate momenta. After systematically
identifying all possible conditions which can be imposed on the parameters such
that different sets of primary constraints arise, we find that the
five-parametric theory space can be compartmentalized into nine different
sectors, based on the presence or absence of primary constraints. This
classification allows to dismiss certain classes of theories as unphysical and
invites further investigations into the remaining sectors, which may contain
phenomenologically interesting modifications of General Relativity.
