An efficient randomized polynomial identity test for noncommutative polynomials given by noncommutative arithmetic circuits remains an open problem. The main bottleneck to applying known techniques is that a noncommutative circuit of size s can compute a polynomial of degree exponential in s with a double-exponential number of nonzero monomials. In this paper, which is a follow-up on our earlier article [AMR16], we report some progress by dealing with two natural subcases (both allow for polynomials of exponential degree and a double exponential number of monomials):

(1) We consider \emph{ + -regular} noncommutative circuits: these are homogeneous noncommutative circuits with the additional property that all the + -gates are layered, and in each + -layer all gates have the same syntactic degree. We give a \emph{white-box} polynomial-time deterministic polynomial identity test for such circuits. Our algorithm combines some new structural results for + -regular circuits with known results for noncommutative ABP identity testing [RS05PIT], rank bound of commutative depth three identities [SS13], and equivalence testing problem for words [Loh15, MSU97, Pla94]. (2) Next, we consider noncommutative circuits: these are noncommutative circuits with layered + -gates such that there are only two layers of + -gates. These + -layers are the output + -gate and linear forms at the bottom layer; between the + -layers the circuit could have any number of gates. We given an efficient randomized \emph{black-box} identity testing problem for circuits. In particular, we show if f \F Z is a nonzero noncommutative polynomial computed by a circuit of size s , then f cannot be a polynomial identity for the matrix algebra M s ( F ) , where the field F is a sufficiently large extension of \F depending on the degree of f .