  We investigate and compare three distinguished geometric measures of
bipartite quantum correlations that have been recently introduced in the
literature: the geometric discord, the measurement-induced geometric discord,
and the discord of response, each one defined according to three contractive
distances on the set of quantum states, namely the trace, Bures, and Hellinger
distances. We establish a set of exact algebraic relations and inequalities
between the different measures. In particular, we show that the geometric
discord and the discord of response based on the Hellinger distance are easy to
compute analytically for all quantum states whenever the reference subsystem is
a qubit. These two measures thus provide the first instance of discords that
are simultaneously fully computable, reliable (since they satisfy all the basic
Axioms that must be obeyed by a proper measure of quantum correlations), and
operationally viable (in terms of state distinguishability). We apply the
general mathematical structure to determine the closest classical-quantum state
of a given state and the maximally quantum-correlated states at fixed global
state purity according to the different distances, as well as a necessary
condition for a channel to be quantumness breaking.
