  Manipulating entanglement, which reflects non-local correlations in a quantum
system and defines the complexity of describing its wave function, represents
the extremely tough challenge in the fields of quantum computing, quantum
information, and condensed matter physics. In this work, by the example of the
well-structured Dicke states we demonstrate that the complexity of these
real-valued wave functions can be accurately tuned by introducing a random-sign
structure, which allows us to explore the regime of the volume-law
entanglement. Importantly, setting nontrivial sign structure one can increase
the entanglement entropy of the Dicke state to the values that are close to
Page's estimates for Haar-random states. The practical realization of these
random-sign Dicke states is possible on different physical platforms with
shallow quantum circuits. On the level of the measurements the change in the
quantum state complexity due to sign structure can be traced out with the
dissimilarity measure that estimates multi-scale variety of patterns in
bit-string arrays.
