In (Finite Fields Their Appl. 46, 38–56 2017), Wu et al. defined the notion of quasi-multiplicative (QM) equivalence among permutation polynomials. Other than showing thoroughly, there is no efficient approach to determine whether two given permutation polynomials are QM equivalent or not. This paper provides new results to determine QM equivalence among permutation polynomials. Based on these, we explicitly provide a list of known permutation polynomials which are equivalent. It is worth noting that a lot of well-known permutation polynomials are QM equivalent, and thus we have only a handful of permutation polynomials which are novel.
