Faithful information transmission through a MAC is possible within its capacity region, which was characterized by Ahlswede 2 and Liao 3 in terms of a so-called single-letter formula, i.e., an entropic optimization problem of fixed bounded dimension that is in principle computable. In network communication settings, the simplest model is a multiple access channel (MAC), where two spatially separated senders aim to transmit individual messages to a single receiver. The central object in our work is a multiple access channel N G defined in terms of a nonlocal game \(G=(\,n)\) in the limit d → ∞, yet they are strictly bounded away from it for any fixed finite d.Information theory is the mathematical theory of communication and signal processing pioneered by Shannon 1. At the same time, entanglement assistance can push the achievable information transmission rates of MACs beyond the classical limit, paving the way for harnessing entangled resources in hybrid classical-quantum information networks. Our findings imply that even for the arguably simplest network information-theoretic setting of a MAC, there is no general solution to the problem of efficiently determining its unassisted communication capabilities, highlighting the need for practical approximation algorithms. Finally, we prove in the unassisted communication setting that it is NP-hard to determine which rates can be achieved for a given MAC. We also show that it is generally undecidable to determine whether the maximal rate pair can be achieved for a MAC with finite-dimensional entanglement strategies. Second, we exhibit examples of channels for which an unbounded amount of entanglement is needed to achieve the maximal possible increase of the achievable rate region. We demonstrate this by constructing a family of classical MACs with surprisingly rich behavior: First, we show that entanglement shared between the senders can strictly increase the capacity region of a classical MAC, proving that entanglement can help in a purely classical communication scenario. Moreover, even unassisted classical MACs exhibit far more complex behavior than previously widely appreciated. In this work, we show that MACs behave in a fundamentally different way in the presence of entanglement assistance, in contrast to the single-sender-single-receiver scenario. However, certain tasks such as classical single-sender-single-receiver communication receive no advantage from entanglement assistance 6. In quantum information theory, communication tasks can be enhanced dramatically if the communicating parties are given access to quantum resources such as shared entanglement 4, 5. Information theory is the mathematical theory of communication and signal processing pioneered by Shannon 1.