Within a histories-framework for quantum field theory, the condition of \bold{persistence of zero} (PoZ for short) tries to capture (a part of) the elusive idea that no cause can act outside its future lightcone. The PoZ condition, however, does not easily carry over to theories like gravity where the causal structure is not only dynamical but indefinite (subject to quantum fluctuations). Despite this, I will suggest how to give PoZ meaning in the causal set...