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The entanglement detection via local measurements can be experimentally implemented. Based on mutually unbiased measurements and general symmetric informationally complete positive-operator-valued measures, we present separability criteria for bipartite quantum states, which, by theoretical analysis, are stronger than the related existing criteria via these measurements. A detailed example is presented to show that the separability criteria can be more efficient than some well-known criteria such as positive partial transpose (Peres-Horodecki) criteria, realignment (computable cross norm) criteria, the criteria based on the realignment of