We study property testing in the subcube conditional model introduced by Bhattacharyya and Chakraborty (2017). We obtain the first equivalence test for $n$-dimensional distributions with a query complexity of $\tilde{O}(n/\boldsymbol{\varepsilon}^2)$, improving the previously known bound of $\tilde{O}(n^2/\boldsymbol{\varepsilon}^2)$.
This result is extended to general finite alphabets with logarithmic cost in the alphabet size. By exploiting the specific structure of the queries we use (which are more restrictive than general subcube queries), we achieve a cubic improvement over the best-known test for distributions over $\{1,\ldots,N\}$ under the interval querying model of Canonne, Ron, and Servedio (2015), attaining a query complexity of $\tilde{O}((\log N)/\boldsymbol{\varepsilon}^2)$, which for fixed $\boldsymbol{\varepsilon}$ almost matches the known lower bound of $\boldsymbol{\Omega}((\log N)/\log\log N)$.
We also derive a product test for $n$-dimensional distributions with $\tilde{O}(n / \boldsymbol{\varepsilon}^2)$ queries and provide a lower bound of $\boldsymbol{\Omega}(\sqrt{n} / \boldsymbol{\varepsilon}^2)$ for this property.