How do polyphosphoric acids differ from metaphosphoric acids in molecular structure?

Polyphosphoric acids and metaphosphoric acids are both types of polymeric phosphoric acids formed through the condensation of phosphoric acid molecules, but they differ significantly in their molecular architecture:

• Polyphosphoric acids consist of a homologous series of long-chain molecules. These chains are characterized by the general formula $\text{H}({\text{HPO}}_3{)}_n\text{OH}$, where $n$ represents the number of phosphorus atoms in the chain ($n=2,3,4,\dots$). The simplest member of this series is diphosphoric (or pyrophosphoric) acid, ${\text{H}}_4{\text{P}}_2{\text{O}}_7$.12
• Metaphosphoric acids are distinguished by their cyclic molecular structures. They are represented by the general formula $({\text{HPO}}_3{)}_n$, where $n=3,4,5,\dots$. The term is also used to describe a viscous, sticky substance that exists as a mixture of both long-chain and ring-shaped forms of $({\text{HPO}}_3{)}_n$.13

Both classes of compounds are formed by the interaction of phosphoric acid molecules, typically at high temperatures, which results in the loss of water and the creation of $\text{P–O–P}$ linkages.14