We prove several results which, together with prior work, provide a nearly-complete picture of the relationships among classical communication complexity classes between $P$ and $PSPACE$, short of proving lower bounds against classes for which no explicit lower bounds were already known. Our article also serves as an up-to-date survey on the state of structural communication complexity.

Among our new results we show that $MA\not\subseteq ZPP^{NP[1]}$, that is, Merlin--Arthur proof systems cannot be simulated by zero-sided error randomized protocols with one $NP$ query. Here the class $ZPP^{NP[1]}$ has the property that generalizing it in the slightest ways would make it contain $AM\cap coAM$, for which it is notoriously open to prove any explicit lower bounds. We also prove that $US\not\subseteq ZPP^{NP[1]}$, where $US$ is the class whose canonically complete problem is the variant of set-disjointness where yes-instances are uniquely intersecting. We also prove that $US\not\subseteq coDP$, where $DP$ is the class of differences of two $NP$ sets. Finally, we explore an intriguing open issue: are rank-$1$ matrices inherently more powerful than rectangles in communication complexity? We prove a new separation concerning $PP$ that sheds light on this issue and strengthens some previously known separations.

We prove several results which, together with prior work, provide a nearly-complete picture of the relationships among classical communication complexity classes between $P$ and $PSPACE$, short of proving lower bounds against classes for which no explicit lower bounds were already known. Our article also serves as an up-to-date survey on the state of structural communication complexity.

Among our new results we show that $MA\not\subseteq ZPP^{NP[1]}$, that is, Merlin--Arthur proof systems cannot be simulated by zero-sided error randomized protocols with one $NP$ query. Here the class $ZPP^{NP[1]}$ has the property that generalizing it in the slightest ways would make it contain $AM\cap coAM$, for which it is notoriously open to prove any explicit lower bounds. We also prove that $US\not\subseteq ZPP^{NP[1]}$, where $US$ is the class whose canonically complete problem is the variant of set-disjointness where yes-instances are uniquely intersecting. We also prove that $US\not\subseteq coDP$, where $DP$ is the class of differences of two $NP$ sets. Finally, we explore an intriguing open issue: are rank-$1$ matrices inherently more powerful than rectangles in communication complexity? We prove a new separation concerning $PP$ that sheds light on this issue and strengthens some previously known separations.