  Quantum discord is a measure of quantum correlations beyond the
entanglement-separability paradigm. It is conceptualized by using the von
Neumann entropy as a measure of disorder. We introduce a class of quantum
correlation measures as differences between total and classical correlations,
in a shared quantum state, in terms of the sandwiched relative R\'{e}nyi and
Tsallis entropies. We compare our results with those obtained by using the
traditional relative entropies. We find that the measures satisfy all the
plausible axioms for quantum correlations. We evaluate the measures for shared
pure as well as paradigmatic classes of mixed states. We show that the measures
can faithfully detect the quantum critical point in the transverse quantum
Ising model and find that they can be used to remove an unquieting feature of
nearest-neighbor quantum discord in this respect. Furthermore, the measures
provide better finite-size scaling exponents of the quantum critical point than
the ones for other known order parameters, including entanglement and
information-theoretic measures of quantum correlations.
