Monotone Classes Beyond VNP.

We study the natural monotone analogues of various equivalent definitions of VPSPACE: a well studied class (Poizat '08, Koiran-Perifel '09, Malod '11, Mahajan-Rao '13) that is believed to be larger than VNP. We show an exponential separation between the monotone version of Poizat's definition, and monotone VNP. We also show that unlike their non-monotone counterparts, these monotone analogues are not equivalent, with exponential separations in some cases. The primary motivation behind our work is to understand the monotone complexity of transparent polynomials, a concept that was recently introduced by Hrube\v{s} and Yehudayoff (2021). In that context, we are able to show that transparent polynomials of large sparsity are hard for the monotone analogues of all known definitions of VPSPACE, except for the one due to Poizat.