We present constant-round interactive proof systems for sufficiently uniform versions of AC0[2] and NC1.
Both proof systems are doubly-efficient, and offer a better trade-off between the round complexity and the total communication than the work of Reingold, Rothblum, and Rothblum (STOC, 2016).
Our proof system for AC0[2] supports a more relaxed notion of uniformity and offers a better trade-off between the number of rounds and the round complexity that our proof system for NC1.
We observe that all three aforementioned systems yield constant-round doubly-efficient proof systems for the All-Pairs Shortest Paths problem.

Correcting a typo in Cor 2 and adding a reference to [11].

We present constant-round interactive proof systems for sufficiently uniform versions of AC0[2] and NC1.
Both proof systems are doubly-efficient, and offer a better trade-off between the round complexity and the total communication than
the work of Reingold, Rothblum, and Rothblum (STOC, 2016).
Our proof system for AC0[2] supports a more relaxed notion of uniformity and offers
a better trade-off between the number of rounds and the round complexity that our proof system for NC1.
We observe that all three aforementioned systems yield constant-round doubly-efficient proof systems for the All-Pairs Shortest Paths problem.