  We develop a novel necessary condition of quantum correlation. It is utilized
to construct $d$-level bipartite Bell-type inequality which is strongly
resistant to noise and requires only analyses of $O(d)$ measurement outcomes
compared to the previous result $O(d^{2})$. Remarkably, a connection between
the arbitrary high-dimensional bipartite Bell-type inequality and entanglement
witnesses is found. Through the necessary condition of quantum correlation, we
propose that the witness operators to detect truly multipartite entanglement
for a generalized Greenberger-Horne-Zeilinger (GHZ) state with two local
measurement settings and a four-qubit singlet state with three settings.
Moreover, we also propose the first robust entanglement witness to detect
four-level tripartite GHZ state with only two local measurement settings.
