The conceptual tension between the special theory of relativity and quantum mechanics is examined from the point of view of the ontological determinateness of events in Minkowski spacetime. While the standard interpretation of quantum mechanics requires an "open", indeterminate future in which events have "fuzzy" attributes before measurement, special relativity has often been thought to require that the future lobes of light-cones at each point contain only fully determinate events. Recent attempts at introducing indeterminateness in Minkowski spacetime are discussed in light of the philosophical implications of experiments violating Bell's inequalities.