If implemented well, Algolia doesn’t have any direct positive nor negative impact on SEO. But there are multiple small indirect positive impacts, due to the fact that search engines prefer websites with a good UX.
1/ Your site's reputation
Search engines have many ways to evaluate if your users like your site (how frequently they type your name, if they re-type a query 5 minutes after they clicked on your site link for the same query…). By providing fast, relevant results and improving the UX of your site, you’ll make your users happier, and this will benefit your search-engine reputation.
Search engines take into account the speed of a website in their rankings. This makes Algolia better than our competitors, since speed is our trademark.
By using our search analytics, and in particular the "queries that returned no result" feature, you can improve your content and catalogue to lower your future bounce-rate.