Quantum and Classical Memory for Storage and Retrieval of Quantum Measurements

We study the persistent quantum and classical memory required for exact heralded storage and retrieval of quantum measurements. Given only black-box access to the measurement channel, we construct an exact compression of the port-based program into an outcome-type record and a quantum register, with total dimension polynomial in the query count. For every universal retrieval processor we prove the bounds $Q+C\ge\log_2 n$ and $2Q+C\ge\log_2\bigl((n-1)d^2+1\bigr)$ on the numbers $Q$ of qubits and $C$ of bits of the program, for $n$-outcome POVMs on $\mathbb{C}^d$ with $d\ge2$. For $n\ge2$, no finite classical memory removes the need for quantum memory: any such processor requires a quantum program of dimension at least $d$, at that dimension its classical alphabet must have at least $n$ symbols, and both minima are attained by a single-query protocol with success $1/d^2$. Combining direct-program compression with a converse from approximate measurement programming gives logarithmic total-memory scaling in $1/(1-p)$ as the uniform success probability $p$ approaches one, at fixed $d,n\ge2$. At fixed input dimension and target success below one, the optimal total-memory scaling is logarithmic in $n$. We also construct self-contained programs under branch-decomposition access and compare the quantum and classical costs of all constructions, including where the outcome-type record is kept: for POVMs with $n$ nonzero rank-one effects the direct program admits a purely quantum allocation whose register is never larger than that of the minimal-Naimark unitary program and which succeeds more often whenever $n>d$.