  Recent work has introduced the "Quantum-Computation Classical-Communication"
(QCCC) (Chung et. al.) setting for cryptography. There has been some evidence
that One Way Puzzles (OWPuzz) are the natural central cryptographic primitive
for this setting (Khurana and Tomer). For a primitive to be considered central
it should have several characteristics. It should be well behaved (which for
this paper we will think of as having amplification, combiners, and universal
constructions); it should be implied by a wide variety of other primitives; and
it should be equivalent to some class of useful primitives. We present
combiners, correctness and security amplification, and a universal construction
for OWPuzz. Our proof of security amplification uses a new and cleaner version
construction of EFI from OWPuzz (in comparison to the result of Khurana and
Tomer) that generalizes to weak OWPuzz and is the most technically involved
section of the paper. It was previously known that OWPuzz are implied by other
primitives of interest including commitments, symmetric key encryption, one way
state generators (OWSG), and therefore pseudorandom states (PRS). However we
are able to rule out OWPuzz's equivalence to many of these primitives by
showing a black box separation between general OWPuzz and a restricted class of
OWPuzz (those with efficient verification, which we call EV-OWPuzz). We then
show that EV-OWPuzz are also implied by most of these primitives, which
separates them from OWPuzz as well. This separation also separates extending
PRS from highly compressing PRS answering an open question of Ananth et. al.
