  We study the effects of quenched disorder in a class of quantum chains with
(p+1)-multispin interactions exhibiting a free fermionic spectrum, paying
special attention to the case p=2. Depending if disorder couples to (i) all the
couplings or just to (ii) some of them, we have two distinct physical
scenarios. In case (i), we find that the transitions of the model are governed
by a universal infinite-randomness critical point surrounded by quantum
Griffiths phases similarly as happens to the random transverse-field Ising
chain. In case (ii), we find that quenched disorder becomes an irrelevant
perturbation: the clean critical behavior is stable and Griffiths phases are
absent. Beyond the perturbative regime, disorder stabilizes a line of
finite-randomness critical points (with nonuniversal critical exponents), that
ends in a multicritical point of infinite-randomness type. In that case,
quantum Griffiths phases also appear surrounding the finite-disorder transition
point. We have characterized the correlation functions and the low-temperature
thermodynamics of these chains. Our results are derived from a strong-disorder
renormalization-group technique and from finite-size scaling analysis of the
spectral gap computed exactly (up to sizes ~10^{7}) via an efficient new
numerical method recently introduced in the literature [Phys. Rev. B 104,
174206 (2021)].
