Web searches often originate from conversations in which people engage before they perform a search. Therefore, conversations can be a valuable source of context with which to support the search process. We investigate whether spoken input from conversations can be used as a context to improve query auto-completion. We model the temporal dynamics of the spoken conversational context preceding queries and use these models to re-rank the query auto-completion suggestions. Data were collected from a controlled experiment and comprised conversations among 12 participant pairs conversing about movies or traveling. Search query logs during the conversations were recorded and temporally associated with the conversations. We compared the effects of spoken conversational input in four conditions: a control condition without contextualization; an experimental condition with the model using search query logs; an experimental condition with the model using spoken conversational input; and an experimental condition with the model using both search query logs and spoken conversational input. We show the advantage of combining the spoken conversational context with the Web-search context for improved retrieval performance. Our results suggest that spoken conversations provide a rich context for supporting information searches beyond current user-modeling approaches.