Chiral α-tertiary amines and related azacycles are sought-after compounds for drug development. Despite progress in the catalytic asymmetric construction of aza-quaternary stereocentres, enantioselective synthesis of multifunctional α-tertiary amines remains underdeveloped. Enantioenriched α-disubstituted α-ethynylamines are attractive synthons for constructing chiral α-tertiary amines and azacycles, but methods for their catalytic enantioselective synthesis need to be expanded. Here we describe an enantioselective asymmetric Cu(I)-catalysed propargylic amination (ACPA) of simple ketone-derived propargylic carbonates to give both α-dialkylated and α-alkyl-α-aryl α-tertiary ethynylamines. Sterically confined pyridinebisoxazoline (PYBOX) ligands, with a C4 shielding group and relaying groups, play a key role in achieving excellent enantioselectivity. The syntheses of quaternary 2,5-dihydropyrroles, dihydroquinines, dihydrobenzoquinolines and dihydroquinolino[1,2-α]quinolines are reported, and the synthetic value is further demonstrated by the enantioselective catalytic total synthesis of a selective multi-target β-secretase inhibitor. Enantioselective Cu-catalysed propargylic substitutions with O- and C-centred nucleophiles are also realized, further demonstrating the potential of the PYBOX ligand.