  Quantum entanglement plays a key role in quantum computation and quantum
information processing. It is of great significance to find efficient and
experimentally friend separability criteria to detect entanglement. In this
paper, we firstly propose two easily used entanglement criteria based on matrix
moments. The first entanglement criterion only uses the first two realignment
moments of a density matrix. The second entanglement criterion is based on the
moments related to the partially transposed matrix. By detailed examples we
illustrate the effectiveness of these criteria in detecting entanglement.
Moreover, we provide an experimentally measurable lower bound of concurrence
based on these moments. Finally, we present both bipartite and genuine
tripartite entanglement measures based on the moments of the reduced states. By
detailed examples, we show that our entanglement measures characterize the
quantum entanglement in a more fine ways than the existing measures.
