Abstract  We study the variances of the coordinates of anevent considered as quantum observables in aPoincare-covariant theory. The starting point is theirdescription in terms of a covariantpositive-operator-valued measure on Minkowski space-time. Besides theusual uncertainty relations, we find strongerinequalities involving the mass and the center-of-massangular momentum of the object that defines the event.We suggest that these inequalities may help toclarify some of the arguments which have been given infavor of a gravitational quantum limit to the accuracyof time and space measurements.