  We study the largest-angle twisted homo-bilayer (LA-THB) systems, hosting
Moir\'eless quasi-crystal (QC) structure. We propose to use these materials to
generate high-angular-momentum (HAM) topological superconductivities (TSCs)
protected by their QC symmetries absent on conventional crystalline materials.
This proposal is based on our universal Ginzburg-Landau theory based analysis
which yields the general conclusion that, when each $D_n$-symmetric ($n$ is
even) monolayer hosts SC with pairing angular momentum $l\le \frac{n}{2}$, the
interlayer Josephson coupling will induce SC with pairing angular momentum
$L=l$ or $L=n-l$ in the LA-THB, determined by microscopic details. The latter
one is just the HAM TSC if $l>0$. Based on our revised perturbational-band
theory, we develop general microscopic framework to study the QC LA-THBs
involving electron-electron interactions, adopting which we study three
examples, i.e. the 30$^\circ$- twisted bilayer graphene, the 30$^\circ$-
twisted bilayer BC$_3$, and the 45$^\circ$- twisted bilayer cuprates. The
$g+ig$- $h+ih$- and $d+id$- TSCs with HAM $L=4,5$ and $2$ can emerge in certain
doping regimes in these systems, respectively.

Superconductivity