  We prove general de Finetti type theorems for classical and free
independence. The de Finetti type theorems work for all non-easy quantum
groups, which generalize a recent work of Banica, Curran and Speicher. We
determine maximal distributional symmetries which means the corresponding de
Finetti type theorem fails if a sequence of random variables satisfy more
symmetry relations other than the maximal one. In addition, we define Boolean
quantum semigroups in analogous to the easy quantum groups, by universal
conditions on matrix coordinate generators and an orthogonal projection. Then,
we show a general de Finetti type theorem for Boolean independence.
