Query auto-completion (QAC) facilitates user query composition by suggesting queries given query prefix inputs. In 2014, global users of Yahoo! Search saved more than 50% keystrokes when submitting English queries by selecting suggestions of QAC. Users' preference of queries can be inferred during user-QAC interactions, such as dwelling on suggestion lists for a long time without selecting query suggestions ranked at the top. However, the wealth of such implicit negative feedback has not been exploited for designing QAC models. Most existing QAC models rank suggested queries for given prefixes based on certain relevance scores. We take the initiative towards studying implicit negative feedback during user-QAC interactions. This motivates re-designing QAC in the more general "(static) relevance-(adaptive) implicit negative feedback" framework. We propose a novel adaptive model adaQAC that adapts query auto-completion to users' implicit negative feedback towards unselected query suggestions. We collect user-QAC interaction data and perform large-scale experiments. Empirical results show that implicit negative feedback significantly and consistently boosts the accuracy of the investigated static QAC models that only rely on relevance scores. Our work compellingly makes a key point: QAC should be designed in a more general framework for adapting to implicit negative feedback.