  We provide a novel criterion for identifying quantum correlation, which
allows us to find connections between Bell type inequalities, entanglement
detection, and correlation. We utilize the criterion to construct witness
operators that can detect genuine multi-qubit entanglement with fewer local
measurements. The connection between identifications of quantum correlation and
Mermin's inequality is discussed. Detection of genuine four-level tripartite
entanglement with two local measurement settings is shown in the same manner.
Further, through the criterion of quantum correlation, we derive a new Bell
inequality for arbitrary high-dimensional bipartite systems, which requires
fewer analyses of the measured outcomes.
