  We introduce a high-dimensional cubical complex, for any dimension t>0, and
apply it to the design of quantum locally testable codes. Our complex is a
natural generalization of the constructions by Panteleev and Kalachev and by
Dinur et. al of a square complex (case t=2), which have been applied to the
design of classical locally testable codes (LTC) and quantum low-density parity
check codes (qLDPC) respectively.
  We turn the geometric (cubical) complex into a chain complex by relying on
constant-sized local codes $h_1,\ldots,h_t$ as gadgets. A recent result of
Panteleev and Kalachev on existence of tuples of codes that are product
expanding enables us to prove lower bounds on the cycle and co-cycle expansion
of our chain complex.
  For t=4 our construction gives a new family of "almost-good" quantum LTCs --
with constant relative rate, inverse-polylogarithmic relative distance and
soundness, and constant-size parity checks. Both the distance of the quantum
code and its local testability are proven directly from the cycle and co-cycle
expansion of our chain complex.
