# Better Protocol for XOR Game using Communication Protocol and Nonlocal Boxes

## TL;DR

This paper introduces a new method for deriving lower bounds on classical communication complexity for XOR games by leveraging nonlocal boxes and existing protocols, generalizing previous bounds with improved intuition.

## Contribution

It extends Buhrman's approach to include a broader class of lower bounds, explicitly constructs protocols using nonlocal boxes, and simplifies the derivation of Linial and Shraibman's bounds.

## Key findings

- Derived a class of lower bounds for classical communication complexity.
- Explicit protocol construction using nonlocal boxes.
- Generalized and simplified previous bounds with improved intuition.

## Abstract

Buhrman showed that an efficient communication protocol implies a reliable XOR game protocol. This idea rederives Linial and Shraibman's lower bounds of communication complexity, which was derived by using factorization norms, with worse constant factor in much more intuitive way. In this work, we improve and generalize Buhrman's idea, and obtain a class of lower bounds for classical communication complexity including an exact Linial and Shraibman's lower bound as a special case. In the proof, we explicitly construct a protocol for XOR game from a classical communication protocol by using a concept of nonlocal boxes and Paw{\l}owski et al.'s elegant protocol, which was used for showing the violation of information causality in superquantum theories.