Blue fish, red fish, live fish, dead fish

We show that the DAG-based consensus protocol Tusk (Danezis, Kokoris-Kogias, Sonnino, and Spiegelman [EuroSys ’22]) does not achieve liveness, under reasonable assumptions which are consistent with its specification. In addition, we give a simple 2-round variation of Tusk with lower latency and strong liveness properties, but with suboptimal resilience. We also show that another 2-round protocol, GradedDAG (Dai, Zhang, Xiao, Yue, Xie, and Jin [Cryptology ePrint Archive, 2024]), also has liveness problems analogous to Tusk.