  Let $A$ be a non-commutative, non-unital $\mathrm{C}^\ast$-algebra. Given a
set of commuting positive elements in the corona algebra $Q(A)$, we study some
obstructions to the existence of a commutative lifting of such set to the
multiplier algebra $M(A)$. Our focus are the obstructions caused by the size of
the collection we want to lift. It is known that no obstacles show up when
lifting a countable family of commuting projections, or of pairwise orthogonal
positive elements. However, this is not the case for larger collections. We
prove in fact that for every primitive, non-unital, $\sigma$-unital
$\mathrm{C}^\ast$-algebra $A$, there exists an uncountable set of pairwise
orthogonal positive elements in $Q(A)$ such that no uncountable subset of it
can be lifted to a set of commuting elements of $M(A)$. Moreover, the positive
elements in $Q(A)$ can be chosen to be projections if $A$ has real rank zero.
