In Shu and Lin, 2011, exponential time (actually polynomial space) algorithms for determining the weak detectability and weak periodic detectability of nondeterministic discrete event systems (DESs) are designed. In this paper, we prove that the problems of determining the weak detectability and weak periodic detectability of deterministic DESs are both PSPACE-hard (hence PSPACE-complete). Then as corollaries, the problems of determining the weak detectability and weak periodic detectability of nondeterministic DESs are also PSPACE-hard (hence PSPACE-complete). (C) 2017 Elsevier Ltd. All rights reserved.