  Quantum coherence and quantum correlations are of fundamental and practical
significance for the development of quantum mechanics.They are also
cornerstones of quantum computation and quantum communication theory. Searching
physically meaningful and mathematically rigorous quantifiers of them are
long-standing concerns of the community of quantum information science, and
various faithful measures have been introduced so far. We review in this paper
the measures of discordlike quantum correlations for bipartite and multipartite
systems, the measures of quantum coherence for any single quantum system, and
their relationship in different settings. Our aim is to provide a full review
about the resource theory of quantum coherence, including its application in
many-body systems, and the discordlike quantum correlations which were defined
based on the various distance measures of states. We discuss the interrelations
between quantum coherence and quantum correlations established in an
operational way, and the fundamental characteristics of quantum coherence such
as their complementarity under different basis sets, their duality with path
information of an interference experiment, their distillation and dilution
under different operations, and some new viewpoints of the superiority of the
quantum algorithms from the perspective of quantum coherence. Additionally, we
review properties of geometric quantum correlations and quantum coherence under
noisy quantum channels. Finally, the main progresses for the study of quantum
correlations and quantum coherence in the relativistic settings are reviewed.
All these results provide an overview for the conceptual implications and basic
connections of quantum coherence, quantum correlations, and their potential
applications in various related subjects of physics.
