  We extend the Hilbert-Schmidt (square norm) distance, previously used to
define the geometric quantum discord, to define also geometric quantifiers of
total and classical correlations. We then compare the dynamics of geometric and
entropic quantifiers of the different kinds of correlations in a non-Markovian
open two-qubit system under local dephasing. We find that qualitative
differences occur only for quantum discords. This is taken to imply that
geometric and entropic discords are not, in general, equivalent in describing
the dynamics of quantum correlations. We then show that also geometric and
entropic quantifiers of total correlations present qualitative disagreements in
the state space. This aspect indicates that the differences found for quantum
discord are not attributable to a different separation, introduced by each
measure, between the quantum and classical parts of correlations. Finally, we
find that the Hilbert-Schmidt distance formally coincides with a symmetrized
form of linear relative entropy.
