  We study genuine tripartite entanglement and multipartite entanglement of
arbitrary $n$-partite quantum states by using the representations with
generalized Pauli operators of a density matrices. While the usual Bloch
representation of a density matrix uses three types of generators in the
special unitary Lie algebra $\mathfrak{su}(d)$, the representation with
generalized Pauli operators has one uniformed type of generators and it
simplifies computation. In this paper, we take the advantage of this simplicity
to derive useful and operational criteria to detect genuine tripartite
entanglement. We also obtain a sufficient criterion to detect entanglement for
multipartite quantum states in arbitrary dimensions. The new method can detect
more entangled states than previous methods as backed by detailed examples.
