  We investigate the rigidity for the Hopf algebra ${\rm QSym}$ of
quasisymmetric functions with respect to the monomial, the fundamental and the
quasisymmetric Schur basis, respectively. By establishing some combinatorial
properties of the posets of compositions arising from the analogous Pieri rules
for quasisymmetric functions, we show that ${\rm QSym}$ is rigid as an algebra
with respect to the quasisymmetric Schur basis, and rigid as a coalgebra with
respect to the monomial and the quasisymmetric Schur basis, respectively. The
natural actions of reversal, complement and transpose of the labelling
compositions lead to some nontrivial graded (co)algebra automorphisms of ${\rm
QSym}$. We prove that the linear maps induced by the three actions are
precisely the only nontrivial graded algebra automorphisms that take the
fundamental basis into itself. Furthermore, the complement map on the labels
gives the unique nontrivial graded coalgebra automorphism preserving the
fundamental basis, while the reversal map on the labels gives the unique
nontrivial graded algebra automorphism preserving the monomial basis.
Therefore, ${\rm QSym}$ is rigid as a Hopf algebra with respect to the monomial
and the quasisymmetric Schur basis.
