  Algorithmic contract design is a new frontier in the intersection of
economics and computation, with combinatorial contracts being a core problem in
this domain. A central model within combinatorial contracts explores a setting
where a principal delegates the execution of a task, which can either succeed
or fail, to an agent. The agent can choose any subset among a given set of
costly actions, where every subset is associated with a success probability.
The principal incentivizes the agent through a contract that specifies the
payment upon success of the task.
  A natural setting of interest is one with submodular success probabilities.
It is known that finding the optimal contract for the principal is
$\mathsf{NP}$-hard, but the hardness result is derived from the hardness of
demand queries. A major open problem is whether the hardness arises solely from
the hardness of demand queries, or if the complexity lies within the optimal
contract problem itself. In other words: does the problem retain its hardness,
even when provided access to a demand oracle? We resolve this question in the
affirmative, showing that any algorithm that computes the optimal contract for
submodular success probabilities requires an exponential number of demand
queries, thus settling the query complexity problem.
