How to find the set of edges for the directed graph associated with a partial order?
As Yury already mentioned, the output size can be too large to hope for subquadratic time, when measured as a function of the input size $n$. But even when the output size is small, very little can be done. In particular, suppose that the input is a partial order with a single comparable pair, chosen uniformly at random among all such partial orders. Then the output size is $O(1)$ but nevertheless it takes $\Omega(n^2)$ queries to find the comparable pair. This is true regardless of whether you're considering only deterministic algorithms or whether you're doing expected case analysis of randomized algorithms.