Autoxidation by sequential peroxy radical hydrogen shifts (H-shifts) and O2 additions has recently emerged as a promising mechanism for the rapid formation of highly oxidized, low-volatility organic compounds in the atmosphere. A key prerequisite for autoxidation is that the H-shifts of the initial peroxy radicals formed by, e.g., OH or O3 oxidation are fast enough to compete with bimolecular sink reactions. In most atmospheric conditions, these restrict the lifetime of peroxy radicals to be on the order of seconds. We have systematically investigated all potentially important (nonmethyl, sterically unhindered) H-shifts of all four peroxy radicals formed in the ozonolysis of α-pinene using density functional (ωB97XD) and coupled cluster [CCSD(T)-F12] theory. In contrast to the related but chemically simpler cyclohexene ozonolysis system, none of the calculated H-shifts have rate constants above 1 s(-1) at 298 K, and most are below 0.01 s(-1). The low rate constants are connected to the presence of the strained cyclobutyl ring in the α-pinene-derived peroxy radicals, which hinders H-shifts both from and across the ring. For autoxidation to yield the experimentally observed highly oxidized products in the α-pinene ozonolysis system, additional ring-opening reaction mechanisms breaking the cyclobutyl ring are therefore needed. We further investigate possible uni- and bimolecular pathways for opening the cyclobutyl ring in the α-pinene ozonolysis system.