QCSP on semicomplete digraphs

We study the (non-uniform) quantified constraint satisfaction problem QCSP(H) as H ranges over semicomplete digraphs. We obtain a complexity-theoretic trichotomy: QCSP(H) is either in P, is NP-complete or is Pspace-complete. The largest part of our work is the algebraic classification of precisely which semicompletes enjoy only essentially unary polymorphisms, which is combinatorially interesting in its own right