  The Clifford hierarchy is a set of gates that appears in the theory of
fault-tolerant quantum computation, but its precise structure remains elusive.
We give a complete characterization of the diagonal gates in the Clifford
hierarchy for prime-dimensional qudits. They turn out to be $p^{m}$-th roots of
unity raised to polynomial functions of the basis state to which they are
applied, and we determine which level of the Clifford hierarchy a given gate
sits in based on $m$ and the degree of the polynomial.
