  We study the notion of $k$-stabilizer universal quantum state, that is, an
$n$-qubit quantum state, such that it is possible to induce any stabilizer
state on any $k$ qubits, by using only local operations and classical
communications. These states generalize the notion of $k$-pairable states
introduced by Bravyi et al., and can be studied from a combinatorial
perspective using graph states and $k$-vertex-minor universal graphs. First, we
demonstrate the existence of $k$-stabilizer universal graph states that are
optimal in size with $n=\Theta(k^2)$ qubits. We also provide parameters for
which a random graph state on $\Theta(k^2)$ qubits is $k$-stabilizer universal
with high probability. Our second contribution consists of two explicit
constructions of $k$-stabilizer universal graph states on $n = O(k^4)$ qubits.
Both rely upon the incidence graph of the projective plane over a finite field
$\mathbb{F}_q$. This provides a major improvement over the previously known
explicit construction of $k$-pairable graph states with $n = O(2^{3k})$,
bringing forth a new and potentially powerful family of multipartite quantum
resources.
