We define a new non-sequential logic element: the random flip-flop (RFF) and show how it can be realized experimentally by using quantum randomness present in a photonic system. By definition RFF operates similarly to the conventional flip-flop except that its clock input functions with probability of 1/2 and otherwise the flip-flop does nothing. Even though the shift from deterministic to random flip-flop may seem minimal, the latter shows an unexpectedly rich behavior and possible areas of application because now not only logic states but also time becomes important. We discuss several digital and analog applications of the novel device in cryptography, over-Turing computing, randomness preserving frequency division, random frequency synthesis and random pulse computing.