Grid workflow systems aim to support large-scale complex e-science and e-business processes. Due to the complexity of these kinds of processes, to control the execution of them in terms of time in the grid workflow execution environments, multiple fixed-date temporal constraints often need to be set simultaneously. Hence the dependency problem between them must be considered and emphasised, which affects the verification effectiveness and efficiency of the fixed-date temporal constraints. However, current relevant workflow verification works do not take into consideration this dependency. Therefore, in this paper, we explore it in depth and analyse its impact on the verification of the fixed-date temporal constraints. Furthermore, based on the temporal dependency, we develop some new temporal verification methods and algorithms. The comparison and evaluation show that these new methods and algorithms make the verification more effective and efficient. All these analyses, new methods and algorithms further strengthen the grid workflow time management.