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We study the largest-angle twisted homo-bilayer (LA-THB) systems, hostingMoir\'eless quasi-crystal (QC) structure. We propose to use these materials togenerate 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 analysiswhich yields the general conclusion that, when each $D_n$-symmetric ($n$ iseven) monolayer hosts SC with pairing angular momentum $l\le \frac{n}{2}$, theinterlayer Josephson coupling will induce SC with pairing angular momentum$L=l$ or $L=n-l$ in the LA-THB, determined by microscopic details. The latterone is just the HAM TSC if $l>0$. Based on our revised perturbational-bandtheory, we develop general microscopic framework to study the QC LA-THBsinvolving electron-electron interactions, adopting which we study threeexamples, 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 certaindoping regimes in these systems, respectively.

High-Angular-Momentum Topological Superconductivity in the Largest-Angle Twisted Homo-bilayer Systems