## The Chasm at Depth Four, and Tensor Rank: Old results, new insights

In an apriori surprising result, Raz (2010) showed that for any $n$ and $d$, such that $\omega(1) \leq d \leq O\left(\frac{\log n}{\log\log n}\right)$, constructing explicit tensors $T:[n]^d \rightarrow \mathbb{F}$ of high enough rank would imply superpolynomial lower bounds for arithmetic formulas over the field $\mathbb{F}$. Using the additional structure we obtain from our proof of the depth reduction for arithmetic formulas, we give a new and arguably simpler proof of this connection. We also extend this result for homogeneous formulas to show that, in fact, the connection holds for any $d$ such that $\omega(1) \leq d \leq n^{o(1)}$.