New 't Hooft anomalies of discrete, higher-form symmetries have been widely studied and shown to give new constraints on the IR physics of gauge theories in the path integral (i.e. Lagrangian) formulation of quantum field theory. In this work, we instead find these anomalies explicitly in the Hamiltonian formulation and derive useful new constraints on the ground states of such models. Our work covers the explicit examples of mixed anomalies involving the 1-form centre symmetry in Yang-Mills, Supersymmetric Yang-Mills, and QCD(adj). I will show how these anomalies appear as non-trivial operator algebras of the symmetry generators. I will then use these algebras to find lower bounds on the ground state degeneracies of these theories, including proving complete spontaneous breaking of chiral symmetries in SU(N) theories.