  By local measurements on party $A$ of a system $AB$ and classical
communication between its two parties, one can achieve a nonlocal advantage of
quantum coherence (NAQC) on party $B$. For the $l_1$ norm of coherence and the
relative entropy of coherence, we generalized the framework of NAQC for two
qubits and derived the criteria which capture NAQC in the $(d\times
d)$-dimensional states when $d$ is a power of a prime. We also presented a new
framework for formulating NAQC, and showed through explicit examples its
capacity on capturing the NAQC states. Moreover, we proved that any bipartite
state with NAQC is quantum entangled, thus the obtained criteria can also be
used as an entanglement witness.
