Alternating cyclic symmetry quadrilinear decomposition (ACSQLD), has been developed as a new algorithm for the processing of four-way data in this work.Owing to its unique optimizing approach, ACSQLD is much more efficient than four-way parallel factor analysis (four-way PARAFAC) in terms of being insensitive to excess number of components and fast convergence in calculation.Moreover, the third-order advantage is investigated through comparing the performances of four-way algorithms (four-way PARAFAC and ACSQLD) with those of two popular three-way algorithms (three-way PARAFAC and self-weighted alternating trilinear decomposition) using simulated data.