It is my understanding that to denote, say, the set of all integer sequences (set of all functions from $\mathbb{N}\to\mathbb{Z}$) we write:
$$\mathbb{Z}^\mathbb{N}\equiv\{f\mid f:\mathbb{N}\to\mathbb{Z}\}$$
What if I want to denote the set of all ordered pairs with elements from $\mathbb{Z}^\mathbb{N}$? That is how do I denote the Cartesian square of $\mathbb{Z}^\mathbb{N}$? Is it this:
$$\left(\mathbb{Z}^\mathbb{N}\right)^2$$
or can I write it like this: $\mathbb{Z}^\mathbb{2N}$? maybe this: $\mathbb{Z}^\mathbb{N\times2}$?
The "Cartesian square" is nothing more than the Cartesian product of a set with itself.
$$\left(\mathbb Z^\mathbb{N}\right)^2$$
is okay, but doesn't look nice.
$$\mathbb Z^\mathbb{N} \times \mathbb Z^\mathbb{N}$$
will be understood by everyone.
The other versions can be misinterpreted for something else completely, don't use them. You are not working with powers of real numbers, don't forget that.