  We show that the phenomenon of quantum contextuality can be used to certify
lower bounds on the dimension accessed by the measurement devices. To prove
this, we derive bounds for different dimensions and scenarios of the simplest
noncontextuality inequalities. The resulting dimension witnesses work
independently of the prepared quantum state. Our constructions are robust
against noise and imperfections, and we show that a recent experiment can be
viewed as an implementation of a state-independent quantum dimension witness.
