# Depth-Independent Lower bounds on the Communication Complexity of Read-Once Boolean Formulas

## TL;DR

This paper establishes fundamental lower bounds on the communication complexity of read-once Boolean formulas, showing that both randomized and quantum protocols require significant communication regardless of formula depth.

## Contribution

It provides depth-independent lower bounds on the randomized and quantum communication complexity of read-once Boolean formulas, extending previous depth-dependent results.

## Key findings

- Randomized complexity lower bound: Ω(√n)
- Quantum complexity lower bound: Ω(n^{1/4})
- Embedding Disjointness problem in formulas for proofs

## Abstract

We show lower bounds of $\Omega(\sqrt{n})$ and $\Omega(n^{1/4})$ on the randomized and quantum communication complexity, respectively, of all $n$-variable read-once Boolean formulas. Our results complement the recent lower bound of $\Omega(n/8^d)$ by Leonardos and Saks and $\Omega(n/2^{\Omega(d\log d)})$ by Jayram, Kopparty and Raghavendra for randomized communication complexity of read-once Boolean formulas with depth $d$. We obtain our result by "embedding" either the Disjointness problem or its complement in any given read-once Boolean formula.