Classification of a specific problem Is it known that $\mathsf{IP}\notin\mathsf{NP}^{cc}\cup\mathsf{coNP}^{cc}$ where $\mathsf{IP}$ is inner product communication complexity problem? Where is the classification of $\mathsf{IP}$ currently in?
Let $\mathsf{coIP}$ denote the problem of returning the negation of the inner product. It has roughly the same complexity as $\mathsf{IP}$ since we can add to both players an extra $1$ bit. It is known that the deterministic communication complexity of a decision problem is upper bounded by the product of the non-deterministic complexities of the problem and its complement. Since $\mathsf{IP}$ has deterministic complexity $\Omega(n)$, it follows that the non-deterministic complexity of $\mathsf{IP}$ and $\mathsf{coIP}$ is $\Omega(\sqrt{n})$, and in particular $\mathsf{IP} \notin \mathsf{NP}^{cc} \cup \mathsf{coNP}^{cc}$.

Sun, Wang and Yu showed that the non-deterministic communication complexity of $\mathsf{IP}$ is in fact $\Omega(n)$, even under the promise that the answer is either $0$ or $1$ (as an integer value).

In the related $\mathsf{MA}$ model, in which we demand perfect completeness but only constant soundness, there is an $O(\sqrt{n}\log n)$ protocol due to Aaronson and Wigderson almost matching Klauck's $\Omega(\sqrt{n})$ lower bound; see for example Pitassi's lecture notes.