Domain filtering local consistencies, such as inverse consistencies, that only delete values and do not add new constraints are particularly useful in Constraint Program- ming. Although many such consistencies for binary con- straints have been proposed and evaluated, the situation with non-binary constraints is quite different. Only very recently have domain filtering consistencies stronger than GAC started to attract interest. Following this line of re- search, we define a number of strong inverse consisten- cies for non-binary constraints and compare their pruning power. We show that three of these consistencies are equiva- lent to maxRPC in binary CSPs while another is equivalent to PIC. We also describe a generic algorithm for inverse consistencies in non-binary CSPs and show how it can be instantiated to enforce some of the proposed consistencies. Finally, we make a preliminary empirical study that demon- strates the potential of strong inverse consistencies.