  In the classic scoring rule setting, a principal incentivizes an agent to
truthfully report their probabilistic belief about some future outcome. This
paper addresses the situation when this private belief, rather than a classical
probability distribution, is instead a quantum mixed state. In the resulting
quantum scoring rule setting, the principal chooses both a scoring function and
a measurement function, and the agent responds with their reported density
matrix. Several characterizations of quantum scoring rules are presented, which
reveal a familiar structure based on convex analysis. Spectral scores, where
the measurement function is given by the spectral decomposition of the reported
density matrix, have particularly elegant structure and connect to quantum
information theory. Turning to property elicitation, eigenvectors of the belief
are elicitable, whereas eigenvalues and entropy have maximal elicitation
complexity. The paper concludes with a discussion of other quantum information
elicitation settings and connections to the literature.
