  Blind delegation protocols allow a client to delegate a computation to a
server so that the server learns nothing about the input to the computation
apart from its size. For the specific case of quantum computation we know that
blind delegation protocols can achieve information-theoretic security. In this
paper we prove, provided certain complexity-theoretic conjectures are true,
that the power of information-theoretically secure blind delegation protocols
for quantum computation (ITS-BQC protocols) is in a number of ways constrained.
In the first part of our paper we provide some indication that ITS-BQC
protocols for delegating $\sf BQP$ computations in which the client and the
server interact only classically are unlikely to exist. We first show that
having such a protocol with $O(n^d)$ bits of classical communication implies
that $\mathsf{BQP} \subset \mathsf{MA/O(n^d)}$. We conjecture that this
containment is unlikely by providing an oracle relative to which $\mathsf{BQP}
\not\subset \mathsf{MA/O(n^d)}$. We then show that if an ITS-BQC protocol
exists with polynomial classical communication and which allows the client to
delegate quantum sampling problems, then there exist non-uniform circuits of
size $2^{n - \mathsf{\Omega}(n/log(n))}$, making polynomially-sized queries to
an $\sf NP^{NP}$ oracle, for computing the permanent of an $n \times n$ matrix.
The second part of our paper concerns ITS-BQC protocols in which the client and
the server engage in one round of quantum communication and then exchange
polynomially many classical messages. First, we provide a complexity-theoretic
upper bound on the types of functions that could be delegated in such a
protocol, namely $\mathsf{QCMA/qpoly \cap coQCMA/qpoly}$. Then, we show that
having such a protocol for delegating $\mathsf{NP}$-hard functions implies
$\mathsf{coNP^{NP^{NP}}} \subseteq \mathsf{NP^{NP^{PromiseQMA}}}$.
