<p>Quantum measurement plays crucial roles in both theoretical foundations and practical applications of quantum information theory. Among these, (<i>N</i>,<i>M</i>)-positive-operator-valued measurements ((<i>N</i>,<i>M</i>)-POVMs) are a class of widely informationally complete measurements. In this paper, we first give the representations of any tripartite and four-partite quantum states by using (<i>N</i>,<i>M</i>)-POVMs and then define special correlation matrices by (<i>N</i>,<i>M</i>)-POVMs. Based on these results, we propose separability criteria under fixed partitions, as well as genuine tripartite and four-partite entanglement criteria for any tripartite and four-partite quantum states. Some examples are provided to illustrate that our criteria are more efficient than the existing ones. Finally, we present an approach to detect entanglement of any multipartite quantum states in any dimensional multipartite systems.</p>