We show an exponential separation between two well-studied models of algebraic computation, namely read-once oblivious algebraic branching programs (ROABPs) and multilinear depth three circuits. In particular we show the following:

1. There exists an explicit n -variate polynomial computable by linear sized multilinear depth three circuits (with only two product gates) such that every ROABP computing it requires 2 ( n ) size.

2. Any multilinear depth three circuit computing IMM n d (the iterated matrix multiplication polynomial formed by multiplying d , n n symbolic matrices) has n ( d ) size. IMM n d can be easily computed by a poly( n d ) sized ROABP.

3. Further, the proof of 2 yields an exponential separation between multilinear depth four and multilinear depth three circuits: There is an explicit n -variate, degree d polynomial computable by a poly( n ) sized multilinear depth four circuit such that any multilinear depth three circuit computing it has size n ( d ) . This improves upon the quasi-polynomial separation of Raz & Yehudayoff (2009) between these two models. The hard polynomial in 1 is constructed using a novel application of expander graphs in conjunction with the evaluation dimension measure (Raz 2006, Forbes & Shpilka 2013), while 2 is proved via a new adaptation of the dimension of the partial derivatives measure of Nisan & Wigderson (1997). Our lower bounds hold over any field.