  We introduce a framework for $\mathbb{Z}$-gradings on cluster algebras (and
their quantum analogues) that are compatible with mutation. To do this, one
chooses the degrees of the (quantum) cluster variables in an initial seed
subject to a compatibility with the initial exchange matrix, and then one
extends this to all cluster variables by mutation. The resulting grading has
the property that every (quantum) cluster variable is homogeneous.
  In the quantum setting, we use this grading framework to give a construction
that behaves somewhat like twisting, in that it produces a new quantum cluster
algebra with the same cluster combinatorics but with different
quasi-commutation relations between the cluster variables.
  We apply these results to show that the quantum Grassmannians admit quantum
cluster algebra structures, as quantizations of the cluster algebra structures
on the classical Grassmannian coordinate ring found by Scott. This is done by
lifting the quantum cluster algebra structure on quantum matrices due to
Gei{\ss}, Leclerc and Schr\"{o}er and completes earlier work of the authors on
the finite-type cases.
