  We define quantum automorphism groups of a wide range of discrete structures.
The central tool for their construction is a generalisation of the
Tannaka-Krein reconstruction theorem. For any direct sum of matrix algebras
$M$, and any concrete unitary 2-category of finite type Hilbert-$M$-bimodules
$\mathcal{C}$, under reasonable conditions, we construct an algebraic quantum
group $\mathbb{G}$ which acts on $M$ by $\alpha$, such that the category of
$\alpha$-equivariant corepresentations of $\mathbb{G}$ on finite type
Hilbert-$M$-bimodules is equivalent to $\mathcal{C}$. Moreover, we explicitly
describe how to get such categories from connected locally finite discrete
structures. As an example, we define the quantum automorphism group of a
quantum Cayley graph.
