Example of a quantifier which is not compatible with ordered pairs.

Question

Let $P(x,y)$ be a binary predicate. I have noticed that the string of quantifiers $\forall x \forall y P(x,y)$ is equivalent to the single quantifier $\forall (x,y) P(x,y)$. Also, $\exists x \exists y P(x,y)$ is equivalent to $\exists (x,y) P(x,y)$. So, I am left to wonder, is there a quantifier $Q$ such that $QxQy P(x,y)$ is not equivalent to $Q(x,y) P(x,y)$? It should not be some sort of exotic quantifier, I am looking for a simple example of a quantifier which is not compatible with ordered pairs.

Answer 1

How about $\exists!$, the unique existence quantifier? Working in the structure $(\mathbb{N};<)$, we have that $\exists !x\exists !y(x>y)$ is true but $\exists !(x,y)(x>y)$ is obviously false.