Cumulative memory---the sum of space used per step over the duration of a computation---is a fine-grained measure of time-space complexity that was introduced to analyze cryptographic applications like password hashing. It is a more accurate cost measure for algorithms that have infrequent spikes in memory usage and are run in environments such as cloud computing that allow dynamic allocation and de-allocation of resources during execution, or when many multiple instances of an algorithm are interleaved in parallel.

We prove the first lower bounds on cumulative memory complexity for both sequential classical computation and quantum circuits.
Moreover, we develop general paradigms for bounding cumulative memory complexity inspired by the standard paradigms for proving time-space tradeoff lower bounds that can only lower bound the maximum space used during an execution.
The resulting lower bounds on cumulative memory that we obtain are just as strong as the best time-space tradeoff lower bounds, which are very often known to be tight.

Although previous results for pebbling and random oracle models have yielded time-space tradeoff lower bounds larger than the cumulative memory complexity, our results show that in general computational models such separations cannot follow from known lower bound techniques and are not true for many functions.

Among many possible applications of our general methods, we show that any classical sorting algorithm with success probability at least $1/\poly(n)$ requires cumulative memory $\tilde \Omega(n^2)$, any classical matrix multiplication algorithm requires cumulative memory $\Omega(n^6/T)$, any quantum sorting circuit requires cumulative memory $\Omega(n^3/T)$, and any quantum circuit that finds $k$ disjoint collisions in a random function requires cumulative memory $\Omega(k^3n/T^2)$.

added more supporting content and cleaned up text and proofs

Cumulative memory---the sum of space used per step over the duration of a computation---is a fine-grained measure of time-space complexity that was introduced to analyze cryptographic applications like password hashing.
It is a more accurate cost measure for algorithms that have infrequent spikes in memory usage and are run in environments such as cloud computing that allow dynamic allocation and de-allocation of resources during execution, or when many multiple instances of an algorithm are interleaved in parallel.

We prove the first lower bounds on cumulative memory complexity for both sequential classical computation and quantum circuits.
Moreover, we develop general paradigms for bounding cumulative memory complexity inspired by the standard paradigms for proving time-space tradeoff lower bounds that can only lower bound the maximum space used during an execution.
The resulting lower bounds
on cumulative memory that we obtain
are just as strong as the best time-space tradeoff lower bounds, which
are very often known to be tight.

Although previous results for pebbling and random oracle models have yielded time-space tradeoff lower bounds larger than the cumulative memory complexity, our
results show that in general computational models such separations cannot follow from known lower bound techniques
and are not true for many functions.

Among many possible applications of our general methods, we show that any classical sorting algorithm with success probability at least $1/\text{poly}(n)$ requires cumulative memory $\tilde \Omega(n^2)$, any classical matrix multiplication algorithm requires cumulative memory $\Omega(n^6/T)$, any quantum sorting circuit requires cumulative memory $\Omega(n^3/T)$, and any quantum circuit that finds $k$ disjoint collisions in a random function requires cumulative memory $\Omega(k^3n/T^2)$.

Added more supporting content, cleaned up text and proofs.

Cumulative memory---the sum of space used over the steps of a computation---is a fine-grained measure of time-space complexity that is a more accurate measure of cost for algorithms with infrequent spikes in memory usage in the context of technologies such as cloud computing that allow dynamic allocation and de-allocation of resources during their execution. We give the first lower bounds on cumulative memory complexity that apply to general sequential classical algorithms. We also prove the first such bounds for bounded-error quantum circuits. Among many possible applications, we show that any classical sorting algorithm with success probability at least $1/\text{poly}(n)$ requires cumulative memory $\tilde \Omega(n^2)$, any classical matrix multiplication algorithm requires cumulative memory $\Omega(n^6/T)$, any quantum sorting circuit requires cumulative memory $\Omega(n^3/T)$, and any quantum circuit that finds $k$ disjoint collisions in a random function requires cumulative memory $\Omega(k^3n/T^2)$. More generally, we present theorems that can be used to convert a wide class of existing time-space tradeoff lower bounds to matching lower bounds on cumulative memory complexity.