  The conditional disclosure of secrets (CDS) primitive is among the simplest
cryptographic settings in which to study the relationship between
communication, randomness, and security. CDS involves two parties, Alice and
Bob, who do not communicate but who wish to reveal a secret $z$ to a referee if
and only if a Boolean function $f$ has $f(x,y)=1$. Alice knows $x,z$, Bob knows
$y$, and the referee knows $x,y$. Recently, a quantum analogue of this
primitive called CDQS was defined and related to $f$-routing, a task studied in
the context of quantum position-verification. CDQS has the same inputs,
outputs, and communication pattern as CDS but allows the use of shared
entanglement and quantum messages. We initiate the systematic study of CDQS,
with the aim of better understanding the relationship between privacy and
quantum resources in the information theoretic setting. We begin by looking for
quantum analogues of results already established in the classical CDS
literature. Doing so we establish a number of basic properties of CDQS,
including lower bounds on entanglement and communication stated in terms of
measures of communication complexity. Because of the close relationship to the
$f$-routing position-verification scheme, our results have relevance to the
security of these schemes.
