The JavaScript Object Notation (JSON) is a popular data format used in document stores to natively support semi-structured data. In this paper, we address the problem of JSON similarity lookup queries: given a query document and a distance threshold $τ$, retrieve all JSON documents that are within $τ$ from the query document. Due to its recursive definition, JSON data are naturally represented as trees. Different from other hierarchical formats such as XML, JSON supports both ordered and unordered sibling collections within a single document. This feature poses a new challenge to the tree model and distance computation. We propose JSON tree, a lossless tree representation of JSON documents, and define the JSON Edit Distance (JEDI), the first edit-based distance measure for JSON documents. We develop an algorithm, called QuickJEDI, for computing JEDI by leveraging a new technique to prune expensive sibling matchings. It outperforms a baseline algorithm by an order of magnitude in runtime. To boost the performance of JSON similarity queries, we introduce an index called JSIM and a highly effective upper bound based on tree sorting. Our algorithm for the upper bound runs in $O(n τ)$ time and $O(n + τ\log n)$ space, which substantially improves the previous best bound of $O(n^2)$ time and $O(n \log n)$ space (where $n$ is the tree size). Our experimental evaluation shows that our solution scales to databases with millions of documents and JSON trees with tens of thousands of nodes.