Quantum logic admits no reasonable material conditional ; any connective that is monotone in a certain technical sense reduces the class of propositions to a Boolean algebra. Consequently, quantum logic struggles to represent the passage of time. One possible workaround is the theory of quantum filtrations developed in the late 1970s and 1980s by Belavkin. It is known, however, that System BV, a deep inference fragment of linear logic that is very close to quantum logic, can handle arbitrary discrete spacetimes.