  The recent development of general quantum resource theories has given a sound
basis for the quantification of useful quantum effects. Nevertheless, the
evaluation of a resource measure can be highly non-trivial, involving an
optimisation that is often intractable analytically or intensive numerically.
In this paper, we describe a general framework that provides quantitative lower
bounds to any resource quantifier that satisfies the essential property of
monotonicity under the corresponding set of free operations. Our framework
relies on projecting all quantum states onto a restricted subset using a fixed
resource non-increasing operation. The resources of the resultant family can
then be evaluated using a simplified optimisation, with the result providing
lower bounds on the resource contents of any state. This approach also reduces
the experimental overhead, requiring only the relevant statistics of the
restricted family of states. We illustrate the application of our framework by
focusing on the resource of multiqubit entanglement and outline applications to
other quantum resources.
