The Blockchain Trilemma Described by a Formula
Taishi Nakai

Akira Sakurai

Shiori Hironaka

Kazuyuki Shudo

Kyoto University
Kyoto, JapanTokyo Institute of Technology
Tokyo, Japan

Kyoto University
Kyoto, Japan

Kyoto University
Kyoto, Japan

Abstract—The blockchain trilemma was first introduced ina 2017 blog post written by Vitalik Buterin, one of the cofounders of Ethereum. This trilemma claims that it is impossible
to simultaneously achieve the three properties of decentralization, scalability, and security in a blockchain. While extensive
analysis of blockchain performance has been conducted, theclaim has been empirically accepted as true. However, no one
has formulated this trilemma mathematically. In this paper, we
present a formula that explicitly expresses the trilemma for Proofof Work blockchains. Furthermore, based on our mathematicalrepresentation of the trilemma, we illustrate the two categories
of approaches for improving blockchain performance under theconstraints of the trilemma.
Index Terms—blockchain, trilemma, decentralization, scalability, security

I. I NTRODUCTION
Blockchain is a distributed ledger that became widely
known when Satoshi Nakamoto published a paper on Bitcoin
in 2008 [1]. Blockchain is characterized by its decentralization and high security. However, compared with traditional
databases, blockchains have lower processing performance,
scalability. There have been many studies to improve its
scalability. For instance, Fast Coin [2] improves scalability
by reducing the block propagation time and SPECTRE [3] byusing a Directed Acyclic Graph for its consensus algorithm.
In addition, sharding, as seen in RapidChain [4], distributes
transactions across multiple groups of nodes, and Rollup [5]processes transactions off-chain to improve scalability. There
are also studies, such as Compact Block Relay [6], that aim toincrease scalability by reducing the size of propagated blocks.
In 2017, Vitalik Buterin, one of the co-founders of
Ethereum [7], wrote in a blog post [8] that a blockchain
can achieve at most two out of the following three properties: decentralization, scalability, and security. This concept isknown as the blockchain trilemma, which is shown in Figure 1.
Buterin defined the three properties as follows:
Decentralization
defined as the system being able to run in a scenario
where each participant only has access to O(c)1 resources, i.e. a regular laptop or small Virtual Private
Server.
Scalability
defined as being able to process O(n) > O(c)transactions.
1 Buterin’s use of the big O notation does not follow its standard definition.
However, out of respect for the original text, we retain the notation O as
presented.

Scalability

Blockchain
Trilemma
Decentralization

Security

Fig. 1. Blockchain Trilemma.

Security
defined as being secure against attackers with up toO(n) resources.
where c denotes the size of computational resources (including
computation, bandwidth, and storage) available to each node,and n denotes to the size of the ecosystem in some abstract
sense. It is assumed that transaction load, state size, and themarket cap of a cryptocurrency are all proportional to n.
Under Buterin’s definition of trilemma, while Bitcoin
achieves decentralization and security, it does not meet thecriteria for scalability. According to a paper that calculated
the maximum transaction throughput for Bitcoin [9], Bitcoin
can only process a maximum of 27 transactions per second.
This is clearly below the number of transactions that a normal
computer can process.
This trilemma has been empirically recognized as accuratewith the increasing analysis of blockchain. However, thetrilemma and its three properties are not well defined, such as
by mathematical formulas, and the current situation is that different definitions are adopted depending on the individual. As
a result, it is unclear whether research on scalability improvements can improve scalability without negatively impacting
decentralization or security. There have been instances where
it has been touted that the constraints of the trilemma havebeen resolved, leading to an improvement in the processing
performance of the blockchain. Moreover, while Buterin’s
definition of the three properties of the trilemma is binary,
more precisely, the three properties have a inverse relationship.
For instance, according to Buterin’s definition, Bitcoin has notachieved scalability. In reality, if you compromise on security
or decentralization to a certain extent, scalability increases
proportionally to the level of compromise. Therefore, Buterin’s
definition does not strictly represent the trilemma.

Proc. 6th IEEE Int'l Conf. on Blockchain(IEEE Blockchain 2023), pp.41-46, December 2023

Total hash rate of minersthat have received the block

In this paper, we demonstrate mathematically a trilemma forblockchains employing Proof of Work by deriving a formulawhere the product of three continuous non-binary quantities,
decentralization, scalability, and security, remains constant. We
also compare the properties of Buterin’s trilemma with theproperties presented in our formula. Furthermore, from ourtrilemma formula, we demonstrate that there are two distinct
approaches to improve the performance of the blockchain
while satisfying the constraints of the trilemma. continuous
non-binary quantities,

1.0
,

0
Elapsed time since the block is generated

II. M ATHEMATICAL D ESCRIPTION OF THE T RILEMMA
We derive a formula that expresses the trilemma, scalability
× security × decentralization = constant, by transforming theequations that express the fork rate and the average blockpropagation time weighted by the hash rate, as defined in
Sakurai et al.’s study [10].
First, we explain the definition of the average block propagation time weighted by the hash rate, Tw . Second, we
represent Tw , which was conveniently expressed as continuous
form in Sakurai et al.’s paper [10], more accurately as the sumof discrete values. Finally, we derive the formula representing
the trilemma from Tw and other equations.
A. Explanation of the Average Block Propagation Time
Weighted by the Hash Rate
We explain the average block propagation time weighted by
the hash rate, Tw , as defined by Sakurai et al. [10]. We firstdescribe the average block propagation time weighted by thehash rate for node i, Tw,i .
Tw,i is the sum of the time it took for a block generated bynode i to reach a certain node, multiplied by that node’s hashrate proportion, summed across all nodes. Figure 2 provides
an example of a graph where the horizontal axis represents
the elapsed time since the block is generated, and the verticalaxis represents the total hash rate of miners that have received
the block up to that time. The average block propagation timeweighted by the hash rate for node i, Tw,i , is represented by
the area of the red shaded region in the graph. Tw,i is expressedby the following equation:
Z ∞Tw,i =−t · u′i (t)dt(1)
0where ui (t) denotes the total proportion of the hash rateof nodes that have not received the block t units of timeafter node i successfully generated the block. In addition,u′i (t) represents the derivative of ui (t). In reality, ui (t) might
not be differentiable, but for simplicity, ui (t) is treated as
differentiable in this context.
Considering the moment each node generates a block, theaverage block propagation time weighted by the hash rate, Tw ,is calculated by taking the weighted average of each node’s
average block propagation time weighted by the hash rate with

Fig. 2. Visualization of Tw,i . Total hash rate of miners that have received
the block before time t versus the elapsed time since the block is generated
by node i. The area of the red shaded region is Tw,i .

respect to each node’s hash rate. Specifically, Tw is expressedby the following equation:
Z ∞
n
X
Tw =
Hi
−t · u′i (t)dt
(2)
0

i=1

where n denotes the number of nodes participating in the
network, and Hi is the proportion of the hash rate that nodei possesses relative to the entire network’s hash rate.
Furthermore, based on the definition of Hi , the following
equation holds.
nX

Hi = 1(3)i=1

B. Representation of the Average Block Propagation Time
Weighted by the Hash Rate, Tw , as a Sum of Discrete Values
We express the average block propagation time weighted
by the hash rate Tw as a sum of discrete values. In the paperof Sakurai et al. [10], Tw was calculated using integration,
assuming the propagation time for each node as a continuousfunction. However, in reality, the propagation time for eachnode is not a continuous function. Therefore, we represent Tw
using discrete values.
When node i generates a block, the time it takes for thatblock to reach all other nodes, weighted by the hash rate of
the receiving node, is defined as the average block propagation
time weighted by the hash rate for node i, Tw,i . If Tij is thetime taken for the block generated by node i to propagate tonode j, then Tw,i can be expressed by the following equation.
In this context, it is assumed that there is no block propagationfrom node i to itself.

Tw,i =nX

Hj Tij(4)

j=1,j̸=i

We derive Tw from Tw,i . The probability of node i generating a block is determined by the ratio of node i’s hashrate to the entire blockchain network’s hash rate. Thus, from

Proc. 6th IEEE Int'l Conf. on Blockchain(IEEE Blockchain 2023), pp.41-46, December 2023

equation (4), the average block propagation time weighted by
the hash rate Tw can be expressed by the following equation.
n
X

Tw =

i=1
n
X

=

Hi Tw,iB = Bh + Btx · ntx

n
X

Hi

i=1

Hj Tij(5)

j=1,j̸=i

C. Derivation of the Trilemma
When the block size is B and the time taken for a block
generated by node i to propagate to node j per byte is tij ,Equation (5) can be represented by the following expression.

Tw =

n
X

Hi

i=1

=B

n
X
i=1

Furthermore, the block size B can be represented using theblock header size Bh , the size of one transaction Btx , and thenumber of transactions contained in one block ntx as follows.

nX

Hj · B · tij

j=1,j̸=in
XHi

Hj tij(6)

j=1,j̸=i

Since equation (6) is a quadratic form (a polynomial consisting only of quadratic terms), it can be represented using
a vector H that describes the distribution of hash rates anda matrix P whose elements represent the block propagation
times between each pair of nodes.
Tw = BH ⊤ P H(7)Specifically, the vector H and the matrix P are represented
as follows.
 H1
 H2  
H= . (8)
 .. 
Hn(12)

By substituting equation (12) into equation (11) and transforming the equation under the condition F ̸= 0, the following
equation can be derived.
Bh + Btx · ntx 1· · H ⊤P H = 1
T
F

(13)

ntx
denotes the number of transactions
In equation (13),Tthat can be processed per unit of time, commonly referred toas Transactions Per Second (TPS). The TPS is often used asa measure of scalability [11]. Thus, the first fraction includes
a representation of scalability when assuming Bh and Btx areconstants and TPS is a metric for scalability. Additionally, ahigh fork rate complicates the decision for miners on whichbranch to adopt. This fragmentation of the hash rate can lead toincreased security risks, such as Double Spending Attack andSelfish Mining [12]. Therefore, the inverse of the fork rate ( F1 )can be interpreted as a representation and metric of security.
If Bh and Btx are constants, and if we consider H ⊤ P H
as a property for decentralization, equation (13) expresses
that the product of the three metrics representing scalability,
security, and decentralization is equal to 1. This implies that itis difficult to improve all three elements simultaneously, which
is precisely the essence of the trilemma. The decentralization
exhibited by H ⊤ P H is discussed in Section III. Bh and Btx ,which are constants here, are also examined in Section V-A.
III. E XAMINATION OF INDICATORS REPRESENTING
DECENTRALIZATION



0
 t21

P = .
 ..

t12
0
..
.

...
...
..
.


t1n
t2n 

.. 
. 

tn1

tn2

...

0

(9)

H ⊤ denotes the transposed vector of H. In the matrix P ,the diagonal elements indicate the time taken for the blockpropagation to itself. Since a block does not propagate between
identical nodes, these diagonal elements are set to 0.
Calculated by Sakurai et al [10], the theoretical fork rate F
can be represented using the block generation interval T andthe average block propagation time weighted by the hash rate
Tw as follows.
F =

Tw
T

(10)

Substitute Equation (7) into Equation (10).
F =BH ⊤ P H
T(11)

We explore the possibility that H ⊤ P H indicates decentralization. First, we show that H represents a property ofdecentralization, and this decentralization can be evaluatedby the variance of the elements of H, Var[H]. Second, we
demonstrate that the decentralization of H indicates throughH ⊤ P H.
A. Decentralization Indicated by H and the metric Var[H]The higher the decentralization, the more evenly distributed
the hash rate is across each node and the greater the numberof nodes constituting the blockchain network as illustrated in
Figure 3. If there is a bias in the hash rates or if the numberof nodes is limited, the benefits of decentralization suchas fault tolerance, attack resistance, and collusion resistancediminish, as discussed in [13]. The vector H represents the
distribution of the hash rate of nodes, and its dimension
inherently indicates the number of nodes n, making it a direct
representation of decentralization.

Proc. 6th IEEE Int'l Conf. on Blockchain(IEEE Blockchain 2023), pp.41-46, December 2023

H ⊤ P H is a quadratic form and can be expressed without
vectors and matrices as:⊤

H PH =nn
X
X

tij Hi Hj .

(17)

i=1 j=1,j̸=iLess decentralizedMore decentralized: nodeThe size of a node represents the amount of the hash rate.

Fig. 3. Decentralization illustrated.

To evaluate the decentralization represented by vector H,
we compute the variance of its elements. The average of theelements of H, denoted as Avg[H], is given by:Pn
Hi
Avg[H] = i=1
n
(14)
1
= .
n
Consequently, the variance of the elements of H, denoted as
Var[H], can be expressed as:Pn
Hi 2− (Avg[H])2
Var[H] = i=1
n
n
(15)
1X 2
1
=
Hi − 2 .
n i=1n
The maximum and minimum values of Var[H] are:
1
1
−
.
(16)
n n2
When all n nodes possess an equal hash rate, implying
the highest decentralization from the perspective of hash rate
distribution, with Hi = n1 (i = 1, . . . , n), Var[H] takes its
minimum value of 0. Conversely, considering the scenario withthe lowest decentralization from the perspective of hash rate
distribution, where only one node in n nodes holds the entire
hash rate of 1 and all other nodes have a hash rate of 0, Var[H]takes its maximum value of n1 − n12 .
Furthermore, under the condition of equation (3), in situations where the number of nodes n is large, implying higher
decentralization, Var[H] becomes smaller. Conversely, whenn is small, indicating lower decentralization, Var[H] becomeslarger.
Thus, the smaller the statistical variance Var[H], the greater
the decentralization of the blockchain. This confirms thatthe vector H represents an element of decentralization, andVar[H] serves as a metric indicating decentralization.
0 ≤ Var[H] ≤B. Decentralization Indicated by H through H ⊤ P H
We aim to demonstrate how the decentralization represented
by H is indicated through H ⊤ P H.

We consider the case where the block propagation time tij
is the same for all nodes. That is, each element of matrix P isconstant at tc . Then, we aim to determine the maximum andminimum values of H ⊤ P H, as well as the vector H that
represents the distribution of hash rates when these maximum
and minimum values are achieved. In this process, we utilize
equation (3) for algebraic manipulations.

H ⊤P H =nn
X
Xtc Hi Hj

i=1 j=1,j̸=i
n
n
X
X

= tc

Hi Hj

i=1 j=1,j̸=i

= tc

n
X



n
X
Hj − Hi 
Hi 

i=1

= tc

n
X

j=1

Hi (1 − Hi )

i=1

= tc

1−

n
X

!
Hi

2

(18)

i=1

From the above equation, the maximum and minimum
values of H ⊤ P H can be expressed as:1.
0 ≤ H ⊤ P H ≤ tc 1 −n(19)When all n nodes have the same hash rate, implying⊤
maximum decentralization, Hi = n1 (i = 1,
 . . . , n), H P Hreaches its maximum value of tc 1 − n1 . If one of the nnodes possesses all the hash rate 1, and all other nodes have ahash rate of 0, implying minimum decentralization, H ⊤ P H
takes its minimum value of 0.
Furthermore, under the conditions of equation (3), when thenumber of nodes n is large, indicating high decentralization,
H ⊤ P H becomes larger. Conversely, when n is small, indicating low decentralization, H ⊤ P H becomes smaller.
Thus, in situations where there is no bias in propagation
speed between nodes, when the blockchain’s decentralization
is high, H ⊤ P H becomes larger, and when the decentralization is low, H ⊤ P H becomes smaller.
In equation (15), Var[H] becomes smaller in highly decentralized situations and larger in less decentralized situations.
On the other hand, H ⊤ P H becomes larger in highly decentralized situationsPand smaller in less decentralized ones.
n
2
This is due to thei=1 Hi part appearing in both equations (15) and (18), and in equation (18), it has a negative

Proc. 6th IEEE Int'l Conf. on Blockchain(IEEE Blockchain 2023), pp.41-46, December 2023

sign. Therefore, the property of H indicating decentralizationis reflected in H ⊤ P H in the same way as Var[H].
In summary, it can be seen that if the block propagation
time tij is the same for all nodes, then H indirectly indicates
decentralization through H ⊤ P H.
For interpretations regarding the situation where the blockpropagation time tij is not equal between each node, see
Section V-B.
IV. C OMPARISON WITH THE T HREE P ROPERTIES OFB UTERIN ’ S T RILEMMA
We compare the properties of the trilemma defined informula (13) with the properties of Buterin’s trilemma. Table
I organizes the three properties of Buterin and those definedin our mathematical formulation.
A. Scalability Comparison
Unlike the existing definition by Buterin, we have adopted
TPS (Transactions Per Second) as the definition of scalability.
This definition provides specific and quantitative information, and is commonly used to measure the scalability of achain [11].
B. Decentralization Comparison
According to Buterin’s definition, if one node controls thehash rate of the entire system, and if each miner can only
access resources of O(c), or if the system is easy to join, thenthis scenario must be rated as highly decentralized. However,this is clearly a centralized state. On the other hand, evenin this scenario, our metric Var[H] increases (indicating ourlower decentralization), making it more appropriate to depict
decentralization.
C. Security Comparison
In Buterin’s article on the trilemma [8], the definitionindicates whether a system is safe against an attacker withresources up to O(n). Our metric, on the other hand, indicatesthe “inverse of the fork rate”, making our definition more
specific and quantitative. In the paper [14], the “ratio of thetotal number of blocks in the main branch to the total numberof confirmed blocks” has already been used as a metric for
security. This metric is close in meaning to the fork rate,
making it common and reasonable to use the inverse of thefork rate as a security indicator.
In conclusion, for the three elements of scalability, decentralization, and security, Vitalik’s definitions are binary andvague. In contrast, our definitions excel in being more specificand quantitative.
V. T WO APPROACHES TO IMPROVE BLOCKCHAINPERFORMANCE UNDER THE CONSTRAINTS OF THE
TRILEMMA
We explore approaches to enhance blockchain performancewithin the constraints of the trilemma. Existing methods
that improve blockchain performance while adhering to thetrilemma’s constraints can be categorized into two types:
those that “reduce Bh and Btx ”, and those that “optimize theelements within P ”.

A. Approaches to Reduce Bh and Btx
In Section II-C, we identified the trilemma within equation(13) by assuming Bh and Btx to be constant. Here, weabandon the assumption of constancy for Bh and Btx inequation (13) and aim to reduce them.
In equation (13), when we reduce Bh and Btx , it does
not adversely affect nTtx , F1 , and the distribution of H. As
a result, the decrease in Bh and Btx can be allocated to
improvements in scalability, security, or decentralization. Thisapproach aims to enhance blockchain’s performance within theconstraints of the trilemma. For instance, Bitcoin’s Compact
Block Relay [6] and the optimization of block generation
notifications in the blockchain using bloom filters [15] improve
blockchain performance by reducing Bh and Btx .
B. Optimization of Elements within P
In this section, we discuss the approaches to improve theperformance of the blockchain by optimizing the elements ofP.
Focusing on Equation (17), by decreasing the value of eachelement in P , we can reduce H ⊤ P H. This is equivalent
to enhancing the bandwidth, reducing latency, and shortening
block verification time. It does not affect the elements that
represent decentralization, which is H. In addition, it does
not decrease nTtx or F1 . Therefore, the reduction in eachelement inside P can be allocated to increase scalability,
security, or decentralization. This is an approach to improve
blockchain performance within the constraints of the trilemma.
For instance, in the selection of neighboring nodes [16], theperformance of the blockchain is improved by prioritizing theselection of nodes with fast block propagation. Technologicalinnovations can enhance communication performance, which
in turn improves the blockchain’s performance, even within
the constraints of the trilemma. Moreover, in Bitcoin’s Compact Block Relay [6], the performance of the blockchain is
improved by sending sketches to related nodes before block
verification is complete.
Looking again at equation (17), especially between nodes
where the product of two hash rates is large, improving communication performance can more efficiently reduce H ⊤ P H
than improving the communication performance between othernodes. In other words, each node should allocate its network
resources to increase bandwidth and reduce latency towards
nodes with larger hash rates. This is equivalent to speeding up
communication between nodes operated by mining pools.
In Section III-B, we assume that the block propagation
time between each node is equal. If the block propagation
time varies between nodes, as demonstrated in this subsection,
H ⊤ P H, which indirectly indicates the decentralization of H,
increase or decrease depending on the block propagation time
between each node.
VI. C ONCLUSION
We represented the trilemma of the blockchain mathematically by deriving a formula where the product of the three

Proc. 6th IEEE Int'l Conf. on Blockchain(IEEE Blockchain 2023), pp.41-46, December 2023

TABLE IC OMPARISON OF T RILEMMA’ S T HREE P ROPERTIES BETWEEN B UTERIN AND O UR M ATHEMATICAL F ORMULATION .
Property
Scalability

Buterin’s Definition
Defined as being able to process O(n)a> O(c)btransactions.

Decentralization

Defined as the system being able to run in a scenario where
each participant only has access to O(c) resources, i.e. a
regular laptop or small Virtual Private Server.
Defined as being secure against attackers with up to O(n)resources.

Security

Our metrics
Number of transactions that can be processed per unittime(TPS).
The abundance of nodes and the lack of bias in the distributionof the hash rate (Variance of hash rate, Var[H]).
Difficulty in causing forks (inverse of fork rate F ).

a n denotes to the size of the ecosystem in some abstract sense. It is assumed that transaction load, state size, and the market cap of a cryptocurrency

are all proportional to n.
b c denotes the size of computational resources (including computation, bandwidth, and storage) available to each node.

continuous quantities: scalability, security, and decentralization, remains constant. Unlike Buterin’s binary definition, thisformula illustrates the inverse relationship among the threecontinuous variables, offering a more precise representation of
the trilemma. We demonstrated that H represents an property
of decentralization, and this decentralization can be evaluatedusing the metric Var[H]. Furthermore, we showed that thedecentralization of H is expressed through H ⊤ P H in thesame manner as Var[H]. We compared the three propertiesof the trilemma as defined by Buterin with those in ourmathematical formula. Additionally, from our formula, wehave shown that there are two categories of approach toenhance the performance of the blockchain while adhering to
the constraints of the trilemma.
In Section V, “Two Approaches to improve blockchain
performance under the constraints of the trilemma”, andspecifically in Section V-B, “Optimization of Elements within
P ”, we mentioned that by improving communication performance, particularly between nodes with a high product of hashrates, the performance of the blockchain can be effectively
enhanced under the constraints of the trilemma. However, byreducing the block propagation time between specific pairsof nodes compared to others, these nodes might gain anadvantage in mining. We will further examine the possibility
that this approach might adversely affect the blockchain in
future research.
While our study focused on blockchains adopting Proof ofWork, we also plan to mathematically represent the trilemma
for blockchains that adopt Proof of Stake.
ACKNOWLEDGMENT
This work was supported by JSPS KAKENHI Grant Number JP21H04872.
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Proc. 6th IEEE Int'l Conf. on Blockchain(IEEE Blockchain 2023), pp.41-46, December 2023