  We study the electronic structure of quasicrystals composed of incommensurate
stacks of atomic layers. We consider two systems: a pair of square lattices
with a relative twist angle of $\theta=45^\circ$ and a pair of hexagonal
lattices with a relative twist angle of $\theta=30^\circ$, with various
interlayer interaction strengths. This constitutes every two-dimensional
bilayer quasicrystal system. We investigate the resonant coupling governing the
quasicrystalline order in each quasicrystal symmetry, and calculate the
quasi-band dispersion. The resonant interaction emerges in bilayer
quasicrystals if all the dominant interlayer interactions occur between the
atomic orbitals that have the same magnetic quantum number. Thus, not only the
quasicrystal composed of the widely studied graphene, but also those composed
of transition metal dichalcogenides will exhibit the quasicrystalline states.
We find that some quasicrystalline states, which are usually obscured by
decoupled monolayer states, are more prominent, i.e., "exposed", in the systems
with strong interlayer interaction. We also show that we can switch the states
between quasicrystalline configuration and its layer components, by turning on
and off the interlayer symmetry.
