In order theory, an antichain (Sperner family/clutter) is a subset of a partially-ordered set, with the property that no two elements are comparable with each other. A maximal antichain is the antichain which is not properly contained in another antichain. Let's take the power set of $\{1,2,\ldots, n\}$ as our partially-ordered set, here the order is given by inclusion. Then my question is, for any given antichain of this partially-orded set, is there any polynomial-time algorithm (with respect to $n$) to verify that this antichain is indeed "maximal"? In other words, verifying that any subset of $\{1,2,\ldots, n\}$ is either contained in, or contains some set from the antichain. Here such algorithm should have polynomial run-time for ANY antichain.
Update: To clarify, here I will treat the size of our antichain as the parameter for the verification algorithm. In other words, my question is: does there exist a verification algorithm, whose run-time is polynomial in $n$ and $m$, where $m$ is the size of the antichain. When the size of our antichain $m$ is exponential in $n$ then such algorithm is trivial (just comparing those elements one by one); but when the given antichain has O(poly(n)) size, this is my interested case. For example, when the antichain is given by $\{\{1\}, \ldots, \{n\}\}$, we certainly do not have to do the brute force comparison.